A Co-development of QuantumMechanics and Lagrangian/HamiltonianClassical Mechanics, with Perspectivesfrom Quantum Electronics and Allied

Fields

Walter Faust (NRL - Retired)

12/02/2012

Contents

1 Introduction 5

2 The Discovery of Quantum Mechanics 6

2.1 The epoch 1900-1923 . . . . . . . . . . . . . . . . . . . . . . . 6

2.2 The transition from classical mechanics to quantum mechanics 6

2.3 1926! Schrödinger presented new concepts which were central,essential, and far-reaching . . . . . . . . . . . . . . . . . . . . 7

2.3.1 The wave function, the wave equation . . . . . . . . . . 7

2.3.2 Operators representing observable parameters belong-ing to the system . . . . . . . . . . . . . . . . . . . . . 8

2.3.3 Especially important features of Schrödinger's wavemechanics . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.4 Concepts that survive from classical mechanics, and some thatnotably do not . . . . . . . . . . . . . . . . . . . . . . . . . . 9

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2.5 Critical features, new and counterintuitive . . . . . . . . . . . 11

2.6 Early observations classically quite unexpected . . . . . . . . . 12

3 Paths in Space: 13

3.1 The confocal laser . . . . . . . . . . . . . . . . . . . . . . . . . 13

3.1.1 The fundamental mode . . . . . . . . . . . . . . . . . . 13

3.1.2 Wavefront curvature . . . . . . . . . . . . . . . . . . . 14

3.1.3 Fabry Perots for light and for matter . . . . . . . . . . 14

3.1.4 Multimode operation . . . . . . . . . . . . . . . . . . . 15

3.1.5 An analogy between atomic energy levels and modesof a resonator . . . . . . . . . . . . . . . . . . . . . . . 16

3.1.6 Eects of gain depletion . . . . . . . . . . . . . . . . . 17

3.2 Merging beams, separating beams; entanglement . . . . . . . . 17

3.2.1 Obvious only in one direction! . . . . . . . . . . . . . . 17

3.2.2 Can superposition be real? . . . . . . . . . . . . . . . . 18

3.2.3 The eikonal ray, classical→ quasi quantum mechanical(Hamilton's work) . . . . . . . . . . . . . . . . . . . . . 19

3.2.4 The eikonal ray, quasi quantum mechanical→ classical(WKB approximation) . . . . . . . . . . . . . . . . . . 19

4 Systems of Quantum Mechanical Notation 19

4.1 Transformations of basis function sets . . . . . . . . . . . . . . 22

4.2 Diagonalization of Hermitian Operators . . . . . . . . . . . . . 23

4.3 Simultaneous Diagonalizability . . . . . . . . . . . . . . . . . . 24

5 A Comparative Exposition on CM and QM 24

5.1 Recollections from classical mechanics . . . . . . . . . . . . . . 24

5.2 Treatments of time-dependence . . . . . . . . . . . . . . . . . 25

5.2.1 Raw Perturbation Theory . . . . . . . . . . . . . . . . 25

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5.2.2 Resonance . . . . . . . . . . . . . . . . . . . . . . . . . 27

5.2.3 Adiabatic approximation . . . . . . . . . . . . . . . . . 27

5.2.4 Born-Oppenheimer approximation . . . . . . . . . . . . 28

5.2.5 Sudden approximation . . . . . . . . . . . . . . . . . . 28

5.2.6 WKB approximation . . . . . . . . . . . . . . . . . . . 28

5.2.7 Born-Oppenheimer approximation; quantum chemicalcalculations . . . . . . . . . . . . . . . . . . . . . . . . 28

5.3 Change/transformation of variables . . . . . . . . . . . . . . . 29

6 Given Two Observables: Their Classical Poisson Bracket andtheir Quantum Mechanical Commutator 29

6.1 The Lagrangian; the classical Poisson bracket . . . . . . . . . 29

6.2 The quantum mechanical commutator of two operators . . . . 31

6.3 The relation between the Poisson backet and the commutator 32

7 Uncertainty Relations 33

8 Consequences of the Uncertainty Principle 35

8.1 Phase Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

8.2 No velocity measurements! . . . . . . . . . . . . . . . . . . . . 35

8.3 The relation between packet velocity and momentum . . . . 36

8.4 Newton's 2nd Law of Motion . . . . . . . . . . . . . . . . . . . 37

8.5 The wave packet yielding the minimal uncertainty-product . . 37

9 Flow of Matter 43

9.1 Conserved Flow of Probability . . . . . . . . . . . . . . . . . . 43

9.2 A ux of free particles encounters a potential wall . . . . . . . 44

9.3 Hamilton's wave equation; the classical eikonal: . . . . . . . . 46

9.4 Hamilton's place in history: . . . . . . . . . . . . . . . . . . . 51

9.5 Quantum mechanical development of the eikonal via the WKBapproximation: . . . . . . . . . . . . . . . . . . . . . . . . . . 52

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10 Angular Momentum 55

10.1 Types of Angular Momentum . . . . . . . . . . . . . . . . . . 55

10.2 Quantization with respect to Rotation . . . . . . . . . . . . . 55

10.3 Experimental Discovery and Theoretical Account for IntrinsicSpin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

10.4 Symmetry upon Exchange of Particles; Fermi and Bose Statistics 57

10.5 Orbital Angular Momentum; Summation over Contributionsto Angular Momentum . . . . . . . . . . . . . . . . . . . . . . 58

10.6 Commuting Operators for Angular Momentum . . . . . . . . . 58

10.7 Corollaries of 10.6 . . . . . . . . . . . . . . . . . . . . . . . . . 59

10.7.1 Commutators connecting the Cartesian coordinates ofthe angular momentum . . . . . . . . . . . . . . . . . . 59

10.7.2 Raising and lowering Jz to complete the set of eigen-functions . . . . . . . . . . . . . . . . . . . . . . . . . . 60

10.7.3 Summing multiple degrees of freedom in angular mo-mentum . . . . . . . . . . . . . . . . . . . . . . . . . . 60

11 The Dirac Relativistic Theory of the Free Particle (Fermistatistics) 61

11.1 Opening sally at the problem . . . . . . . . . . . . . . . . . . 61

11.2 The Dirac equation . . . . . . . . . . . . . . . . . . . . . . . . 63

A The Gaussian Beam; the Fundamental Ray in Free Space 68

B Initial Development of the Lagrangian, Following GCM 68

C The Sturm-Liouville Problem 70

References

[LLQM] Landau and Lifshitz, Quantum Mechanics

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[GCM] Goldstein, Classical Mechanics

[DVQM] Davydov, Quantum Mechanics

[DWIQM] Dicke and Wittke Introductory Quantum Mechanics

[CTQM] Cohen-Tannoudji, Quantum Mechanics

[SQM] Schi, Quantum Mechanics

[DCQM] Dirac, Quantum Mechanics

[BDRQM] Bjorken and Drell, Relativistic Quantum Mechanics

[CS] Condon and Shortley, Theory of Atomic Spectra

[HGTOM] Heine, Group Theory in Quantum Mechanics

[HSRP] Hehre, Radom, Schleyer, and Pople, Ab Initio Molecular OrbitalTheory

1 Introduction

Classical mechanics (CM) is appreciated in a relatively intuitive fashion.Darwin's agencies, over millions of years, have organized our brains to dealwith this matter eciently as a lady robin knows how to build a nest withouthaving previously seen her mother build one.

But microscopes and accelerators are very recent inventions, so QM (quan-tum mechanics) and SR (special relativity) don't come so naturally; GRremains a mystery to most of us (general relativity; black holes represent aneven more recent discovery). GCM, incidentally, remarks that QM requiresa much more violent recasting than does SR. These recastings are foundnecessary, respectively, as material objects get tiny and/or move very fast(and nally, in intense gravitational elds).

We benet from a succession of surges in human understanding, apparentsuccesses. Regarding the associated brilliance: The innate gifts, the labors,of those who have made the great strides certainly are not to be minimized.But one's chancy arrival at a particularly productive perspective deservesemphasis as well; and this suggests humility for most of us.

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2 The Discovery of Quantum Mechanics

2.1 The epoch 1900-1923

Historically, the entry into QM came through insights re the corpuscularnature of light: Planck's treatment of black-body distribution (1900); Ein-stein's treatment of the photoelectric eect (1905); and Compton's treatmentof x-ray scattering from atoms (1923).

2.2 The transition from classical mechanics to quantummechanics

We had not fully arrived at QM until a wavelength was associated with thepropagation of material particles they emulate light. De Broglie (1924) rst

associated wavelike properties, hence a wavelength λ =h

p, with motion of

material objects (p the momentum).

The bridge between QM and CM is a narrow one. One indication is thatmany books on the one topic scarcely refer to the other. And it developsthat the applicable skills, the methodologies, are quite dierent.

From a modern perspective, Erwin Schrödinger may seem to have faced astaggering challenge akin to reverse engineering: to construct a microscopicmodel on the basis of limited microscopic information. The macroscopicmodel of CM had already been brought to maturity, by Lagrange (1788) andHamilton (1833), but in retrospect it was quite far removed from the ultimatemicroscopic model of QM. The microscopic model should not have seemedunique, but far from it.

It is now appreciated that the assertions that ow, for a given dynamicalsystem, from a valid QM model become indistinguishable from those of CMin the limit (from the CM view) that the actionS becomes large relative toPlanck's constant h ' 6.6 × 10−27erg · sec ; or (from the QM view) in thelimit of large quantum #'s. (Terms will be further dened as we progress).But this point, taken alone, aords little insight into Schrödinger's problem.

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Viewing from a perspective more supportive of Schrödinger, indications hadarisen earlier, that the motion of matter manifests the character of wavepropagation, hence that that there is an associated wavelength.

Reports in this line from Wikipedia include:

The Old Quantum Theory of Bohr and Sommereld (1913/25) had laid outintegral numbers of wavelengths for an electron tracing a Kepplerian orbitabout a nucleus, accounting for discrete electronic energy levels.

de Broglie (1923/4) associated the wavelength of a free particle with momen-

tum λ =h

p:

Following Planck's E = hν and Einstein's E = mc2, de Broglie equatedthe two but with mc2 replaced, for material particles, by mv2 to obtain

hν = mv2. Replacing ν byν

λ, (again adopting v instead of c for the velocity

of a material particle), he obtainedhν

λ= mv2 or

h

λ= mv = p , rearranged

to the now-familiar λ =h

p.

2.3 1926! Schrödinger presented new concepts whichwere central, essential, and far-reaching

2.3.1 The wave function, the wave equation

Schrödinger published an equation describing how the matter wave shouldevolve in time the matter wave equivalent of Maxwell's equations andused it to derive the energy spectrum of hydrogen. We present this here,without further preparation, as:

i~∂

∂tψ (q, t) = − ~2

2m∇2ψ (q, t) + V (q, t) (1)

The wave function ψ of a system is a complex-number function (conveyingamplitude and phase) of the specic state in which the system is found at agiven instant. Thus

√−1 = i and complex algebra enter. Phase naturally

enters into the wave function for wave propagation or for the evolution of

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an entangled system, localized or an extended. The sense is similar to thatin electronics.

However, in electronics we write a voltage function of time as V eiωt, thenretain only the real part |V | cos (ωt). QM diers in that any complex quantitygets multiplied by its complex conjugate or with a hermitian, ie a real,operator M sandwiched between to evaluate an observable quantity. Thisagain yields a real number, but thus in a dierent fashion from electronics.

When ψ is an eigenfunction of each of a set of commuting operators M,the state carries as information the corresponding value of each operator.

ρ (q) ≡ ψ (q)∗ ψ (q) denes the probability density at point q, and´ρ (q) dq =

1, with the integral covering the space of variables q. That is, since thesystem must show up somewhere in the space or hyperspace involved (forthe present we assume perfect detectors) the probabilities sum or integrateto unity. To deal with the issue of sum or integrate as the function set isdiscrete or continuous, we introduce a new symbol

ffl to encompass both.

Thus, instead of the simple integral above, we writefflψ† (q)ψ (q) dq = 1 ,

where again the variables q span the space.

Exception: The above normalization is infeasible where (as typically in scat-tering problems) one wishes to describe a steady ux arriving, say along thex-axis from x = −∞ , at an experimental arrangement in the vicinity of x = 0. Instead, one employs typically a unit plane wave, as ψ (x, t) = e(ωt−kx) ;or in three dimensions, ψ (r, t) = e(ωt−k·r). Note that the phase, as chosen,advances with increasing t (new news arrives) and recedes with increasingr (late news hasn't yet arrived).

2.3.2 Operators representing observable parameters belonging tothe system

A set of operators characteristic of the system, such that when each isexercised with ψ, it conveys the value of an observable characteristic of thesystem. More specically, when the operator Λ is applied in ψ†(q)Λψ(q) ,the result represents the value λ at position q . The average over the spaceof q may be expressed as 〈λ〉 =

fflψ†Λψ, where 〈λ〉 is read as The average or

expectation value of operator Λ. (Subsequently, under femtosecond lasers,this is examined yet more critically.) Apart from the position operators

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x, y, z, these operators typically are partial derivatives (Henceforth we use ~

forh

2π)

1. The energy E = i~∂ψ

∂t.

2. The x -component of linear momentum −i~ ∂∂x

.

3. The z-component of angular momentum −i~ ∂

∂ϕz.

2.3.3 Especially important features of Schrödinger's wave me-chanics

There remain other very essential features of the most general state function,the operators, and their interplay, yet to be developed.

1. The wavefunction is burdened with all the information that can beknown about the system, even in principle.Conversely: A function which accommodates every measurable prop-erty of a system is automatically complete in this sense. This is acritical point toward Dirac's 4-component model for the SR electron.

2. Superposition, interference, and entanglement. These are developed atlength, below. Schrödinger evidently understood these quite well, cfhis tale of Schrödinger's Cat (see also Cat further below); he evenemployed the German word for entanglementVershränkung (dic-tionary 1: crossing arms or legs, crossing over, folding, clasping;dictionary 2: interlacing, interconnection).

3. Canonically conjugate observables, and their constraint by the Heisen-berg uncertainty principle.

2.4 Concepts that survive from classical mechanics, andsome that notably do not

A dynamical system (with the term inclusive of photons and other bosonforce-carriers) which invites our study perhaps closed, perhaps subject to

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external inuences; and the state of this system.

Observable variables associated with a state, such as position r , momentump, energy E which, beyond the very constituents of the system, characterizea particular conguration of the system. For canonically conjugate pairs,even these encounter circumscription (ie they are not utterly independent asin CM), due to the principle of uncertainty.

Velocities are not measured in QM, for this would require two measurementsof position nearby in time. The rst would contribute spread to the momen-tum, thus upsetting the second.

Angular momentum nds an important place but displays its own pe-culiar feature in intrinsic spin.

Time, a parameter, simply tick-tocs onward, indierent to particularsystems, until we encounter SR and GR.

The classical action S of a system survives if there is a single pathwhich weighs overwhelmingly (more accurately a tight collection ofneighboring paths). S survives as the phase ϕ of the QM wave func-tion ψ = ρeiϕ. Along any single path, it describes the accumulation ofphase along the arc of propagation of the central ray, in a geometricaloptical picture (this corresponds to the WKB approximation, whichconsiders only the rst order in h). This path and its accumulatingphase ϕ comprise the eikonal, as developed at length below. To re-state: A multiplicity of nearby paths, lying within a suciently smallneighborhood, interfere constructively because the local δ-variation ofthe phase ϕ = S is vanishingly small.

If there are multiple paths to a given nal point, the relative phases ϕ enterin the interference between them (see below).

Prominent among the missing in QM are features of continuity: In CM, forany two feasible values of a system property, any intermediate value alsois typically feasible at least for some limited neighborhood, and barringsingularities, chaos, etc. In many instances of QM energy levels, linear andangular momentum, and ultimately spacetime itself this property is lacking.

Connement in one parameter induces discreteness in another cf Bohr's OldQuantum Theory of the atom, or a particle in a box.

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In CM, rotation about an axis invokes an innite group. QM generally dealsin nite representations of axial rotation, none of which is faithful to theinnite group. And twice around 2π is not equivalent to once around!

2.5 Critical features, new and counterintuitive

Interference: The frequency of visible light (∼ 6x1016Hz ) is so large, andhence λ = c/ν is so small (∼ 5 × 10−7cm) even though c is rather large(∼ 3 × 1010cm/sec) that visualization of interference calls for equipmentnot available on the old farm (eg, no IR nor radio equipment). For massiveparticles, λ is yet much smaller.

Superposition: This principle interplays closely with interference, and is fullyas important. LLQM (p 7) rates this as the chief positive principle of quan-tum mechanics (uncertainty evidently is regarded as negative). If statesΨ1and Ψ2 yield two denite and dierent values of a given variable (i.e. theyare eigenstates) , then the (normalized) sum of the two, ψ+ = 1√

2(ψ1 + ψ2)

will, upon any single measurement, yield exactly the rst or exactly the sec-ond eigenvalue.

Let us suppose that each ofψ1andψ2carry distinct eigenvalues of the sameset of commuting operators, and that each may be taken as a product overthat set. Individual measurements entail a winner-take-all sort of randomcompetition among such product terms, one term only scoring. The termthat scores in an individual measurement will carry its peculiar correlationof the eigenvalues, all belonging exclusively to one or the other of ψ andψ2.The scoring probabilities will respect their complex squares. The term pureis applied to such sum-wavefunctions, as well as to those of a single term (cfmixed, below).

Matter waves: Cited above were the early insights of de Broglie, Schrödinger,and Bohr/Sommerfeld.

Interference of matter waves: Davisson and Germer (1927) electron dirac-tion from a Ni surface; see below. Ever more compelling demonstrationshave since arisen: Today neutron diraction from crystals is standard tech-nique. In Sven Hartmann's billiard ball experiment, entire atoms, fromtwo distinct spatial paths, interfered, as manifested in the emission (Beach,Hartmnn, and Friedberg, Phys. Rev. A25, 2658 (1982). In more recent work,

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interference has been observed even between two colliding aggregates of verylow-pressure gas crystals [Kasevich, Science 298, 15 Nov. 2002, p. 1363].

See below, after Paths in Space have been treated, the extended discussionof entanglement of spatially separated entities, as represented by a super-posed wavefunction following two quite distinct spatial paths.

2.6 Early observations classically quite unexpected

In 1927 at Bell Labs, Davisson and Germer ring slow moving electrons ata crystalline nickel target discovered electron diraction: Distinct waves,scattered from dierent units of the periodic crystalline structure, interferein their distribution over a detector surface; ie they form a pattern of dots astheir relative phases add constructively or cancel. It seems that the electronsare indeed suciently tiny (reference to discussion above), even for modestvoltages.

To restate: In electron diraction from a crystal, there is a multiplicityof emergent wave-streams, each bearing energy and information on thepotential values of observable quantities. Statistically those incident upona given detector-point together describe the potential presence of individualelectrons at that point. Again restating: These streams jointly contributeto the probability that an electron will be detected where they converge at agiven point in the space of the subsequent paths, where a detector is pro-vided. That probability of detection is computed as the complex square ofthe sum of the several amplitudes at that given point. Only the probabilities,not the amplitudes, can be observed directly.

Any such information stream, e.g. the one approaching the Ni surface or theseveral departing is variously known as a wave function Ψ, a probabilityamplitude, a wave vector, a state vector, etc. (These are quite dierentfrom spatial vectors, being similar only in spanning several dimensions oversome set of basis functions. They inhabit a function or Hilbert space,reecting concepts of quasi-length and projection, orthonormality, etc.)Since these streams each carry energy as well as information, they are real;they do not comprise just an artice for calculation of the probabilities.

Simplify momentarily to a case in which there are just two such emergentstreams. It will be found that if either stream is intercepted such as to assessits information content, that nal interference at the subsequent detector

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must be subject to destruction. An amplitude can, in full safety in thisregard, contribute to such an assessment of probability only once, along apath of propagation. These features, counterintuitive as they may seem,have been veried and elaborated in further experiments ranging on to recenttimes. (Aside, re in full safety: Recent experiments have rened this issueto read that there must be at least a chance of such destruction if one isto obtain statistical information of such a nature which chance may beminimized, in a sort of nibbling. Jaws dropping? I'm myself not entirelyclear on this controversial matter. See Science 333, p. 690, 5 Aug., 2011.)

Thus a beam of matter of a given momentum, encountering a periodic struc-ture, behaves much like a monochromatic beam of light encountering, say,a diraction grating. Even for modest momentum or KE values, the wave-length will be very small, as asserted above; and for high voltages, it will besmaller still.

3 Paths in Space:

A wave amplitude may, from its construction, follow a prescribed path, iethis feature is embedded therein. Under this general heading we will considerissues centered about the following three subtopics: A confocal laser thecavity, the lowest mode, and gain depletion; Spatially merging or separatingpaths of propagation; The eikonal ray for a free particle.

3.1 The confocal laser

3.1.1 The fundamental mode

The fundamental mode of a confocal laser models also the most severelyradially-conned axially symmetric optical mode of free space: ie an idealray. See Appendix A for the explicit eld amplitude E(r, z). In microwaveterms it is TEM00n radially a Gaussian packet propagating along the z-axis converging to a waist (a Raleigh waist), then diverging.

Specication of diameter parameters alone would leave adjustable the tight-ness of focus. To quantify this, let the wavefronts conform to two sphericalsurfaces (potentially coinciding with laser mirrors), each having radius of

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curvature b, located symmetrically to each side of the waist at z = ±b/2 ;the center of curvature of each thus lies at the center of the other. The min-

imal spot radius, dening the waist, is w0 =√

bλ2π

; and the waist diameter

is 2w0 =

√2bλ

π. Note that the waist-size is proportional to the geometric

mean of the wavelength λ and the axial length b. Note that the scale of the

diameters is the Fresnel numbera2

bλ.

For green light, λ = 5x10−5cm , and a reasonable b = 102cm , the spotdiameter is 8x10−2cm = 0.8mm.

3.1.2 Wavefront curvature

For the wavefunction of a particle path to express the varying wavefront cur-vatures found essential to the ideal rays of Appendix A, there must be anappropriate correlation among axial position and radial position; phase, andamplitude such as to produce the varying wavefront-normals. The embed-ding of this path in the wavefunction thus cannot be represented in a simpleproduct wavefunction of the nature Ψ1 (x) Φ1 (y). An integral of correlationmust appear as an essential character, so that we have immediately a stateof superposition.

Indeed, note in Appendix A that, with E (r, z) in polar form elnEei·phase wedo have a product, and that the factors dier in their dependencies upon rand z, as promised.

In a recent article, A Dierent Angle on Light Communications, Willner,Wang , and Wong, Science 337, 655, 10 August, 2012, helical wavefronts areillustrated. In these, axial and radial position-variables are joined by θ , sothat phase and amplitude depend upon three coordinate variables. I havecopied pdf the article, with the images, for attachment, as Spiral modes.pdf.

3.1.3 Fabry Perots for light and for matter

Consider again a pair of concave mirrors arranged confocally, as for a laser(but without the gain medium).

Toward green light the mirrors are, say, ~98% reecting, ~1.5% transmitting,with 0.5% loss. Let a monochromatic beam of such light approach externally,

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from the left. If such a setup is created without further specications, themost likely result is the obvious one: that a small fraction of the incidentintensity will be found circulating between the mirrors, and perhaps a mere(0.0152 = 0.0025%) of that intensity will exit the cavity at the opposite end(The exact proportion hinges upon a geometric series... wildly varying phaseswith a strong tendency toward cancellation). However, if the mirror sepa-ration is given a special value very near an integral number of wavelengths,a substantial fraction of the incident energy will exit to the right; we havestruck a Fabry-Perot resonance.

For charge-free materials, ie atoms, etc. (thus excluding space-charge repul-sion), the beam-shape expression given in Appendix A should yet describethe spatial propagation of the center of mass, with a multiplying function onthe internal variables describing their internal dance. Further, a Fabry-Perotdemonstration is possible in principle with a suciently monochromatic par-ticle beam [a very tall order!...very small λ] but compare for diculty withthe Davisson Germer experiment of 1924. See the attached papers Gundyresonant tunneling..., sent me by my former student Prof. Martin Gunder-sen, in reply to this point.

3.1.4 Multimode operation

For two co-travelling waves of distinct but nearby frequencies, the QM ex-pectation energy is just the weighted average of the two hν ′s , but this maynot describe well the detected signal. If the eective duration of the measure-ment is brief (as set by the integration time the detector rise/decay time,or a gate duration), a measurement sequence may exhibit a beat freqency.This actually was the rst experimental result from the rst gas laser, HeNeat 1.15 µm (running simultaneously in a number of distinct modes withinthe Doppler width of the Ne line). Dirac (DCQM, section 27, p. 109) seemsto exclude such a case, implicitly restricting to a wave function of uniqueenergy.

The extreme instance of this lies with femtosecond-pulse lasers. There exista variety of techniques to induce a laser to run in the manner of a short pulsecirculating between two mirrors: Synchronized optical gating; loss in a cellof saturable-absorber dye; and nally use of a prime-mover pulsed laser topump a slave laser of the same length.

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In this last method, one achieves repetitively-pulsed pumping of the slave gainmedium and depletion of its gain by the circulating pulse of the slave. Thesetrim respectively the leading and trailing pulse-edges my own contributionto the art. In each case, the output is a repetitive train of pulses; the Fourierrepresentation of such a train is a uniform comb of frequencies.

A Ti:sapphire laser can support this with gain essentially across the spectrumof green light; correspondingly it may produce pulses of ∼ 5× 10−15sec at arate ∼ 100Mhz . Rather than clocking such pulses with a fast detector, oneis then soon occupied in using them for a clock!

3.1.5 An analogy between atomic energy levels and modes of aresonator

A generic atom and a generic laser cavity are each a sort of resonator, havingcharacteristic frequencies. Considering a) losses from the mirror system or b)loss of excitation from an atom (emission, collisions), the frequencies may betaken to be complex number quantities, real + imaginary (Think of Laplacetransforms. Thanks to Art Schawlow for this viewpoint.)

For the atom or for the laser, the frequencies are largely real. For an atom thereals comprise the energy-level spectrum. The lossy imaginary parts com-monly are relatively small. For a laser, the real parts comprise a decoratedcomb of frequencies. The principal spines of the comb, of TEMq modes,are determined by the longitudinal mode numbers. These TEMqmodes arecircularly symmetric, radially Gaussians. The losses, the imaginary parts, as-sert themselves in the transverse dependence, with the TEMq modes lowestand others ascending quickly from them.

Consider the energy escaping through a circular hole drilled in one mirror asa manner of output (again, my contribution), somewhat smaller than the 1stFresnel zone, on the central axis. Reect upon a progression of increasinghole sizes. See the accompanying gure McCumber.pdf (D. E. McCumber,Bell System Tech. J. 48, 1919, computed and written at my suggestion).Quoting McCumber, The (Huygens scalar) eld amplitude... for a typicalmode can be written in the form f jlp (ρ) e−ilϕ ; thus the loss curves are labeledby the angular index node number l and by the radial node index number ρ. The superscript j on f distinguishes the hole-bearing mirror j = 1 and the2nd mirror j = 2 . Provided that the incremental loss of the fundamental

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l = 0, p = 0 through the hole is less than the excess loss of the next circularlysymmetric mode l = 0, p = 2 , the character of the low-lying modes willremain roughly the same; this is parallel to perturbation theory for energylevels of atoms (See Raw Perturbation Theory, regrettably below).

3.1.6 Eects of gain depletion

This refers to exhaustion of population inversion on a realized or potentiallaser emission line, through some competing process:

First: Continuing the above discussion of increasing coupling-hole size, andMcCumber's gure: When the loss of the fundamental exceeds that of thenext mode l = 1, p = 0 (lower than l = 0, p = 2, not of the same circular sym-metry), laser operation will make the jump, in appreciation of minimal loss;the fundamental will fail from gain depletion, in the course of the competi-tion. The suggestion from perturbation theory, that the shape of the mode isas yet little disturbed, is irrelevant. NB also that the laser prefers the lesserloss, not the greater output energy. Quantitively, if the mirrors' Fresnel # is1.0, this permits ~0.3% coupling. For smaller F#, the discrimination amongmodes increases [MCCumber Fig. 5], but at the price of increasing loss forthe fundamental, l = 1, p = 0 .

Second: In a similar vein, consider a pair of atomic congurations connectedby a number of emission lines and potentially lasers, such as the Ne 3p→ 2ssystem. The preferred lines are likely not strong ones as familiarly observedin spontaneous emission, for this would compete with lasing. Unfortunately,calculation of line strengths for lines which are weak (but important in lasing)is relatively likely to produce large fractional errors.

3.2 Merging beams, separating beams; entanglement

3.2.1 Obvious only in one direction!

Soon after Javan, Bennett, and Herriott had demonstrated the rst gas lasers(1961), Bennett aspired to use a partially reecting mirror to achieve co-travelling beams, merging, from two distinct lasers running on very nearbyfrequencies then to exhibit an interference pattern racing across the face ofa detector (a time-gated image converter). I dissuaded him, asserting that

17

one would thereby simply have constructed a wavefunction of two additivecomponents, initiating in distinct paths, but summed at the detector. Iregarded the result as obvious, viewing optics as an extension of the contextof radio; in fact, natural multimode operation had already manifested therelationship. For matter- waves, it was less obvious, even in 1962!

A full appreciation of path-carrying wavefunctions might have led me (asit did not) to an earlier recognition of entanglement, wherein initially co-local particles or waves separate, subsequent to some point in propagation.This picture holds even if, as the experiment terminates, functions Ψ arelocalized in Rome, and functions Φ in Tokyo: the measured variable valuesemerge correlated according to the terms in ψ. The demonstration which Ihad considered trivial (cf Bennett above) may be construed as a converse todemonstrations of entanglement, which are popularly considered much moreremarkable: entities merging vs. entities separating. Most remarkable is thatthese features apply to particulate matter as well as to photons! Arguablywe should have been prepared for this since de Broglie.

3.2.2 Can superposition be real?

Since Shrödinger's original work, many have felt deeply disturbed by the con-cept of superposition, beginning with Einstein, Podolsky, and Rosen (1935;Wikipedia Schrödinger's Cat, ref. 5). In fact, Schrödinger introduced hiscat as a yes indeed reply to Einstein et al., re Einstein's Spooky action ata distance.

Bohm and others pursued a class of hidden variable theories, essentiallysupposing that the entangled partners somehow knew right along whichterm in the correlation would hold for a given exercise. This instance parallelsthat of the uneducated lady robin of the very rst paragraph of this tome.

John Bell's original theorem on entanglement dealt with a system of twodouble-valued variables, say two spins 1/2, yielding for each magnetic sub-components ±1/2 . The declared result is that that no hidden variable theorycould produce correctly the distribution of measured values.

A relatively recent article in Science dealt with three double-valued variables.Here it was claimed that no hidden-variable theory could produce correctlyeven the result of single exercises of the experiment. [Sorry this is frommemory only; I haven't retrieved the two actual publications, either for two

18

or for three variables. For two variables, however, see N. David Mermin,RMP 65, 803.] Wikipedia characterizes Bell's work as controversial.

3.2.3 The eikonal ray, classical→ quasi quantummechanical (Hamil-ton's work)

Discussion of the eikonal ray, promised above, is deferred, pending an exten-sive (and enabling) review of Lagrangian-Hamiltonian classical mechanics.This conveys a classical account for material wave-propagation along thecentral ray of any single path, the furthest reach of CM toward QM.

3.2.4 The eikonal ray, quasi quantummechanical→ classical (WKBapproximation)

Conversely, a suitable degradation of QM, by exercise of theWKB approximationresurrects the classical picture of the eikonal ray. This is deferred also, forthe same reason.

4 Systems of Quantum Mechanical Notation

There are three systems of notation to be mentioned, that of Schrödinger(the language of dierential equations, wave equations); that of Heisenberg(matrix equations); and that of Dirac (bras to the left and kets to the right,hence brackets). In each we have states, on the one hand, and operators onthe other. The the matter of the adjoint, and its role in yielding exclusivelyreal observable values, is represented somewhat dierently among them.

Doran and Lasenby eqn 8.3 (for instance) uses appropriately a hybrid format,the spin spectrum being discrete (for starters, twofold) and the spatial partbeing continuous; more much later.

The sense of multiplication is perhaps most in need of discussion for matrixnotation (This is ok for nite matrices, but ill-suited for continuous, ratherthan discrete, eigenvalues). State and state-adjoint vectors are treated ascolumn and row vectors, the latter the conjugate of the former. Operators aretreated as square matrices. Multiplication is standard matrix multiplication.

19

There are two important classes of operators (here we conveniently use thelanguage of matrices):

1. Unitary operators can be employed to accomplish a linear transforma-tion from one basis set to another such set (here set pertains to thefunctional components cited above).

2. Hermitian operators correspond to real-number observable quantities.Their eigenvalues are the allowed values which observable quantitiesmay attain in individual exercises, eg, one electron at a time (as op-posed to statistical distributions of outcomes, over many trials).

In general, the inverse of a matrix M of m rows and m columns is givenby another m x m matrix, determined element by element as follows: Thei,j element is evaluated as the determinant of a smaller, (m− 1) × (m− 1)matrix, obtained by the deletion from M of the ith row and the jth column.The resulting m×m is then divided by the determinant of the original m×m.

The adjoint of any of these square matrices is the matrix of transposed com-plex conjugates.

The adjoint of a unitary matrix is equal to the inverse of that matrix; this isessential to preserve ortho-normality with the change of basis set.

The adjoint of a hermitian matrix is equal to the matrix itself,(Hba

)∗= Hab,

ie < (H) is symmetric, and = (H) is antisymmetric across (perpendicular to)the diagonal descending to the right. The diagonal elements are real.

A state is represented by a single-column matrix, listing the the amplitudes(complex #'s) for the functional components of that state in the basis setadopted. The adjoint state, a single-row matrix (or vector), is built of thecomplex conjugates of the components. (In passing column>row and takingthe conjugate, this emulates the adjoint-taking of a matrix.)

Here the components for a state or its conjugate are the function-projectionsof that state onto the members of an an orthonormal complete set in the ap-propriate complete function-space, according to the theory of such spaces.The projection connecting a pair of matrices is given by the integral over thespace (real and/or spin, etc.) of the product of the functions (or perhaps of afunction and an adjoint function), in accordance with the theory of complete

20

sets of functions (See Appendix C re the Sturm-Liouville problem, and/orGoogle Dirichlet).

In the Schrödinger picture, the two-factor product state-adjoint x state,ψ†ψ describes a probability distribution over the component basis set, ie adistribution over the real/spin space (continuous or discrete) of the functionalvariables. Since the sum/integral of probabilities over all congurations mustbe unity, we have

ˆψ† (q)ψ (q) dq = 1. (2)

LLQM p. 6, eqn. 2.1 give as the most general expression of this sort thedouble integral

´ ´Ψ† (q) Ψ (q′)φ (q, q′) dqdq′; but this seems to convey inter-

action at a distance; and it is entirely unfamiliar (?).

Some circumstances may call for density matrices say when the externalparameters themselves cover some classical distribution; accordingly the sys-tem is described by a set of pure wavefunctions ψi, each given a statisticalweight. Each is applied to the operator matrix H as ψ†Hψ, with the weightedsummation being performed only afterward essentially a classical weighting.Whereas the term pure is applied to single or to multiple-term wavefunc-tions without such further post-summation, the corresponding term here ismixed. This topic will not be pursued here at greater length.

The adjoint of a column state-vector is the row vector of complex conjugates.The adjoint H†of the operator H representing an observable (hermitian, self-adjoint) obeys, forΨ,Φ any wave amplitudes, Ψ†HΦ = Φ†H†Ψ = Φ†HΨ =Ψ†(HΦ) = (Ψ†H)Φ , etc.

The value-distributions of observable quantities (or their expectation val-ues) appear as triple products: in the order left-to-right, an adjoint staterow vector, a square hermitian matrix for some observable, and the statecolumn vector itself. The expression

´ψ(q)†ψ(q)dq = 1 above corresponds

to case of the unit matrix operator (solid diagonal 1's). For the present weassume perfect detectors.

In Schrödinger's treatment, a state function is a complex function of observ-able quantities, expandable in some basis set for the function-space. Theadjoint state function is just the conjugate, unless spin-matrices enter. An

21

observable is treated by a specic corresponding operator. An observabledistribution is described as the sum/integral over a sandwich: adjoint statetimes observable operator Ω times state,

´ψ(q)†Ω(q)ψ(q)dq. Each entity

may be a function of the system variables (space/spin) and the time.

Operators may involve partial derivatives as noted earlier. Particularly im-

portant observable operators are, for instance, x and Px = −i~ ∂∂x

, these

representing the x-position of a particle, say, and the x-component of its lin-

ear momentum; also the time t and the energy operator H = i~∂

∂t. Here

~ ≡ h

2π. These choices accommodate symmetries of spacetime. Immediately

we can spot anomalies from the perspective of CM: There appears no massin the momentum, nor a velocity. For a stateΨ ∼ ei(kx−ωt), it will develop

that momentum is represented in a wavelength λ where k =2π

λ; and the

velocity appears as that of a wave, λν =λω

2π.

In Dirac notation, the parallel is again a sandwich called a bracket: bratimes operator times ket. It appears as <variables|operator|variables>. Thevariable set typically are eigenvalues for some set of operators sucient forcompleteness within a function-set.

...

4.1 Transformations of basis function sets

Appropriate unitary operators U (matrices) can be used to perform ro-tations (transform linearly) in function (Hilbert) space through similaritytransformations U−1MU on the functional basis set for the states. Unitaryoperators preserve probabilities.

Not only are the diagonal elements of a hermitian matrix H real, the ba-sis can further be rotated to produce exclusively zeroes o-diagonal todiagonalize H. The corresponding basis functions then are recognized aseigenfunctions of the observable operator H, with the eigenvalues along thediagonal of H.

22

4.2 Diagonalization of Hermitian Operators

Suppose that at the outset we are given such a basis set ϕj, spanningthe pertinent function space but otherwise random, and the correspondingnon-diagonal H . We wish to rotate to a diagonalizing basis set ψi and thecorresponding diagonal Hdiag.

Any eigenfunction ψ (λi) conforms by denition to Hψ (λi) = λiψ (λi). Em-ploying the identity matrix I, write this as (H − λiI)ψ (λi) = 0 . Thesecomprise n equations indexed by i. Since ψi is not zero, the matrix H − λIis null, indeed null as applied to any function within the the space dened bythe set ψi, thus over the space of ϕj. The determinant of a null matrixvanishes, for functions within the allotted space; ie |H − λI| = 0 .

Note that the expression of the zero determinant stands independent of anyparticular basis(!) covering the given function space; the equation actuallydoes not refer to any specic basis. Expansion of the determinant yields annth-order polynomial, which in general will have n roots the solution set λifor the zeroes (a worthy problem unto itself). The set λi then comprisethe diagonal elements of Hdiag , which is thereby known (note that the orderof the λi and the rows/columns is arbitrary).

The span of functionalities available must remain as embedded in the orig-inal set ϕj , which defnes the function space. Hence, in accomplishing theintended rotation, we are dealing with the numerical coecients of expansionof the individual ψi 's in the basis set ϕj.Since, for the n matrix equations (H − λiI)ψ (λi) = 0 , and within thisfunction space, the determinant of the matrix H − λI vanishes, repeatedapplication of this matrix with each of the set λi overdetermines the coef-cients and thus the set ψi. We are free to take one of the original set ϕj,say ϕ1 as the rst member ψ1(λ1) of the new set; thus the rst coecientand only nonzero coecient for ψ1(λ1) is 1.

With the expansion coecients for the other ψ′s taken as a column c ofnumbers (0, c2, c3, ...cj) we write the matrix equation (H − λiI) c = 0 totreat each of the other ψi 6=1 and their λi 6=1. A second index may be addedto the c's to enumerate the λ′s , as 0, c2k, c3k, ...cjk. Restating from the priorparagraph, ψ1 is represented by the column (1, 0, 0, ...). The n columns maybe taken to comprise a matrix Φ, when multiplied by H−λI, gives the matrixof new columns Ψ : (H − λI) Φ = Ψ.

23

For each given λi , this comprises a system of simultaneous equations tocomplete the solution for the coecients describing the corresponding ψi.The rst row, with λ = λ1 , is satised identically through the freedom justexercised, so the solution is essentially (n− 1) × (n− 1) . Elaborating onthis: (1, 0, 0, ...) projects φ1 only onto ψ1 , and the other ϕ′s do not projectonto ψ1.

Like simultaneous equations in general, the system may be solved by multi-plication from the left by the inverse of H − λI , to yield the coecients ofthe transformed function set Ψ in terms of the basis still aorded by Φ.

4.3 Simultaneous Diagonalizability

If two operator matrices, say energy and momentum, commute, the samebasis set will diagonalize them; they are simultaneously diagonalizable.Demonstration: Any pair of matrices which in some basis set are both di-agonal will commute (fairly obvious). Consider rotation with any unitarymatrix per 4.1; recall U−1MU ; commutation of two operators is not aected(again fairly obvious).

Following Schur's Lemma of group theory (HGTQM, p. 100; Appendix D, p.418), the state-energies of a system fall into blocks according the representa-tions of commuting operators with simultaneously diagonal eigenfunctions.Rotations respecting the commuting set only rearrange the internal contentsof those blocks.

5 A Comparative Exposition on CM and QM

5.1 Recollections from classical mechanics

One should have CM adequately in hand before confronting the passage to,or the construction of, QM. Acquaintance is presumed with Lagrangian CM;but do see Appendix B, for it is not trivial.

Do we have at the outset a fully reliable concept of coordinate or of mo-mentum, of canonically conjugate pairs? This entails a number of com-plications that bedevil us in CM, eg constraints holonomic or not (GQM

24

refers to some of these as vicious). For the context of QM the latter issuescarcely arises; we are not dealing with balls rolling upon surfaces, etc.

LLQM emphasize that typically there are external conditions rigidly pre-scribed by materials whose own uctuations are not addressed as QM issues;such are static or resonant electromagnetic elds, gratings, etc. Exceptionsto this point are posed, eg, by ab initio quantum chemical calculations onisolated molecules a la Born-Oppenheimer (HRSP).

5.2 Treatments of time-dependence

The issue of time-dependence in the energy H is handled quite dierently inQM, as opposed to CM.

5.2.1 Raw Perturbation Theory

For H (t) = 0 , such theory enables treatment of a problem not subject toexact solution, on the basis of an initial exact solution of a nearby problem.ForH (t) 6= 0 , it further treats a wide diversity of external inuences.

Following SQM Ch. VII: It is assumed that we have in hand the normalizedeigenfunctions un and eigenvalues En for H0 . If the set un is degenerate, itshould rst be diagonalized in H ′. We write the Hamiltonian H as the sumof two parts H0 and H ′ , where the Schrödinger equation forH0 has alreadybeen solved exactly and H ′ is small, so that it invites expansion in a powerseries.

We replace H ′ by λH ′ and express the perturbed eigenfunctions and eigen-values in terms of the parameter λ . We shall then have in the powers of λthe successive orders of perturbation theory. We assume that the series forthe perturbed ψ and and energy W are analytic in λ between 0 and 1 , andset λ = 1.

ψ = ψ0 + λψ1 + λ2ψ2 + λ3ψ3 + ... (3)

W = W0 + λW1 + λ2W2 + λ3W3 + . . . . (4)

25

Substituting into the wave equation, we obtain an equation for each order ofλ , of which we consider here only the rst two, the zero and the rst orders.The zero order teaches us nothing new:

H0ψ0 = W0ψ0 (5)

H0ψ1 +H ′ψ0 = W0ψ1 +W1ψ0. (6)

We expand ψ1 in the basis set un. That is, ψ1 =fflna

(1)n un. Here ψ0 = um

(note that W0 = Em) is any one of the unperturbed states, but which isspecial in that we have focused upon it, to inquire for its energy shifts due tothe perturbation treated (currently in rst order only) byW1. Again we haveused

fflto mean summation/integration according to discrete/continuous

states.

Substitution into the 1st order equation just above gives

n

a(1)n H0un +H ′um = Em

a(1)n un +W1um (7)

Let u†k, Ek describe any other individual state for which we wish to learn theshift in Em associated with it.

We multiply through by u†k and implement the summaton/integration overthe index n, applying the orthonormality of the basis set.

n

u†ka(1)n H0un +

u†kH

′um = Em

a(1)n u†kun +W1

u†kum . . . (8)

Two cases arise:

1. For the special case k = m , there is a self-shift of energy. Becauseffluk † un = δkn , the rst term is akEk = amEm. The second term

is H ′mm. The third term is amEm. And the fourth is W1. Becausefflu†mun − δmn, we obtain W1 = H ′mm.

26

2. The more general case k 6= m is more interesting. We have

a(1)k Ek +H ′km = a

(1)k Em, (9)

a(1)k =

H ′kmEm − Ek

. (10)

Since the lowest state is often the one of particular interest, indicatingit for the role of state ψm, rewrite this as

a(1)k =

−H ′kmEk − Em

. (11)

The kth state ψk is admixed. This state, is, with this amplitude addedto the state ψ0 = um. If H ′km > 0 , ak is negative; state ψk is addedto ψm with a negative amplitude.

Reversing the perspective, consider exchanging the roles of two particularstates ψm and ψk.

The ultimate results of perturbation theory, in increasing orders, displaystriking eects of avoided crossing and exchange of character of the eigen-functions.

5.2.2 Resonance

Absorption/emission of energy from an EM eld subject to hν ∼ ∆E, somesplitting of levels may build over many cycles. Typically only two energylevels experience such eects, since dephasing expresses itself rapidly for pairseven slightly o-resonance. Typically we do not attempt to re-model theatom/molecule in its entirety with an H (t) inclusive of such dephasing.

In some very recent work with femtosecond light pulses of extreme strength,a new sort of remodeling is indeed found appropriate.

5.2.3 Adiabatic approximation

For Stark splittings, for deection of a beam by an inhomogeneous magneticeld, the eld varies so slowly that the system's energy follows the continuousvariation of each energy level. The characteristic frequencies of the variationare small compared to the level spacings, so that they are quite o-resonsnce.(The usage of adiabatic in thermodynamics is actually derivative from theusage here.)

27

5.2.4 Born-Oppenheimer approximation

For isolated molecules, widely disparate time-scales for electronic and fornuclear motions enable treatment of the electronic changes as adiabatic withrespect to the nuclear motions. With the nuclei taken xed, one computeselectronic energies. With the latter taken as aording a background potential,the nuclear motions are then treated. (See HRSP. But the H2molecule isunfavorable; H presenting a light nucleus, the vibrations do not separatecleanly from electronic energy intervals.)

5.2.5 Sudden approximation

This is essentially opposite to adiabatic: The eigenstates of the initial hamil-tonian, abruptly nding themselves in a new hamiltonian, individually projectonto collections of eigenstates of the new hamiltonian.

5.2.6 WKB approximation

(Wentzel Kramers Brillouin) Here one takes only the rst order only of anexpansion in Planck's constant. This is useful in tunneling problems, wherein an important region the energy is negative. This is employed for alphadecay, the escape of a He4 unit from a nucleus. Presentation of the WKBmethod will be withheld until the semi-classical development of the eikonalhas been completed.

5.2.7 Born-Oppenheimer approximation; quantum chemical cal-culations

[HRSP] These can yield the shape, the energy, and the vibrations of anisolated molecule. Here H is typically independent of time; and this iswidely the case otherwise. Quantum chemical problems don't introduce non-integrability, which lies in a category GCM cited as vicious (see above).This allows transformations of variables to make all the coordinates cyclic(quite a boon; see later development toward the eikonal).

28

5.3 Change/transformation of variables

There remains in common with CM the issue of changes of variables. Thesecan facilitate solutions by discovery of invariants. Also, they can rectify issuesof asymmetry among the independent variables. They are accomplished inCM by Legendre transformations. Corresponding to this in QM are canon-ical transformations accomplished by the unitary operators discussed above.

We must take due warning as given by GCM that, as such transformations areapplied, the characters of the variables recognizable specically as positionsand momenta may be abandoned (eg, angles don't have dimension length).They may evolve to more general coordinates and their time-derivatives. Yettheir status as conjugate pairs will remain; this behavior is delivered by thenature of CM Legendre transformations and again by that of QM unitarytransformations.

6 Given Two Observables: Their Classical Poisson

Bracket and their QuantumMechanical Commutator

Note at the outset: The Poisson bracket of CM is couched in derivatives ofsystem parameters. The commutator of QM is couched in operators, oftenthemselves derivatives, which work upon the system wave function.

6.1 The Lagrangian; the classical Poisson bracket

The physics here may be considered elementary, but the algebra gets a bitintricate. Therefore the initial development is consigned to Appendix B, thisseeming preferable to the pretense that we all are already familiar with thematter.

Path integrals: The Lagrangian L, and its friends (units energy), particularlyH (see below), enter into path integrals over time. These involve variation ofthe independent variables and t , and hence of L. There are two distinct typesof path integrals. Variations δ entail xed end-points (qi, t); the variationsare viewed as virtual. For variations 4, even the end points vary in [qi, t],and t varies as necessary for feasible motions (H conserved).

29

It is central to principles of variation of path integrals that these pairs qi,pibe regarded as independently variable; ie, in Lagrangian computations withL(qi, q

i, t), the nature of q

i as the time derivative of qi is in this sense ignored.

The δ−variation of L is

δL =∑i

∂L∂qi

δqi +∂L

∂qiδqi+

∂L

∂tδt =

∑i

(piδqi + piδqi) +∂L

∂tδt. (12)

Seeking a more fully symmetrical choice, we invoke the Legendre transfor-mation to −H (qi, pi, t) = L (qi, qi, t) − qipi. (nb: In the similar relations ofthermodynamics, the minus sign on the left is not present). In the variation

− δH =∑i

piδqi + piδqi − piδqi − qiδpi+∂L

∂tδt, (13)

the rst and third terms cancel. Thus

δH = −∑i

∂L

∂qiδqi + qiδpi −

∂L

∂t=∑i

(−piδqi + qiδpi)−∂L

∂t, (14)

since∂L

∂qi= pi by Lagrange's equations. We have transformed from L to a

new energy variable H (qi, pi, t). The new CM equations of motion are

∂H

∂qi= −pi, (15)

∂H

∂pi= qi. (16)

We note that these are quite symmetrical. We nd that

∂H

∂t= −∂L

∂tdH

dt=∑i

(∂H

∂qiqi +

∂H

∂pipi

)− ∂L

∂t= −∂L

∂t.

Thus H is independent of time unless there is an explicit dependence thisfeature arises spontaneously.

30

Consider two arbitrary functions of these same variables, f (qi, pi, t) andg (qi, pi, t). The Poisson bracket is dened in general as

[f, g]q,p ≡∑i

(∂f

∂qi

∂g

∂pi− ∂g

∂pi

∂f

∂qi

), (17)

so thatdf

dt=∂f

∂qiqi +

∂f

∂pipi + ∂f/∂t =

∂f

∂t+ [f,H] . (18)

($$ a very important result!). We may recognizedH

dtabove as [H,H]q,p +

∂H

∂t, where this Poisson bracket vanishes. Unless the hamiltonian depends

explicitly upon the time, it is a constant.

...

We assume that we have now in hand, from CM, mature concepts of gener-alized coordinates and momenta, as we approach QM.

We must yet develop further our appreciation of these pairs of independentvariables in QM, where we will recognize them as canonical conjugates,whose pairwise commutators are proportional to H (hence very small). Pois-son brackets will serve as a point of entry on the question How might QMhave been discovered in a more present-day spirit? The reference is to suchvery modern questions as Now, really, is such and such quantity zero (cf.the neutrino mass), or is it just very small? We will return to this pointmuch later [search zero vs. very small], after having discovered the classi-cal version of the Schrödinger Equation.

6.2 The quantum mechanical commutator of two oper-ators

For the corresponding QM development in terms of the commutator , wedraw liberally from LLQM, Ch. 11.

As asserted earlier, The wave function Ψ determines completely the state ofa physical system in quantum mechanics.

31

With q now representing the variable set [qi, pi], we refer to Ψ (q) and toan observable f (q). The concept of a derivative of f (q) with respect totime cannot be dened in quantum mechanics in the same way as in classicalmechanics. (QM delivers only a distribution over probabilities for f .). Wedene

favg (q, t) ≡ˆ

Ψ (q, t)† f (q, t) Ψ (q, t) dq. (19)

Then we dene the time derivative of f as the derivative, with respect totime, of the mean value of f , ie as

dfavgdt

=∂

∂t

ˆΨ (q, t)† f (q, t) Ψ (q, t) dq

=

ˆΨ†∂f

∂tΨdq +

ˆ∂Ψ†

∂tΨdq +

ˆΨ†f

∂Ψ

∂tdq. (20)

Using∂Ψ

∂t=

i

~HΨ, but since H = H†, we have

∂Ψ†

∂t= − i

~(HΨ)† =

− i~

Ψ†H† = − i~

Ψ†H, and

dfavgdt

=

ˆΨ†∂f

∂tΨdq +

i

~

ˆΨ†fHΨdq − i

~

ˆΨ†H†fΨdq

=

ˆΨ†∂f

∂tΨdq +

i

~

ˆΨ†fHΨdq − i

~

ˆΨ†HfΨdq

=

ˆΨ†(∂f

∂t+i

~(fH −Hf)

)Ψdq. (21)

6.3 The relation between the Poisson backet and thecommutator

Comparison to the CM result above $$ shows a correspondence between theCM Poisson bracket and the QM expression with the commutator: [H, f ] vs.(i/H)(Hf − fH). Specically, as we consider systems increasingly classicalin character, the QM expression i (fH −Hf)must evolve to accommodate arole consistent with the CM expression H[Hf − fH]. Relative to the macro-scopic systems described by CM, ~ is extremely small, only 6.6×10−27erg·sec.

32

Here evolve means that the variables/parameters attaining macroscopicmagnitudes come to dwarf ~. This is why QM was not discovered until earlyin the 20th century.

DWIQM point out that the arguments above, applied to the expectationvalue of an arbitrary operator Q , yield

d 〈Q〉dt

=

⟨∂Q

∂T

⟩+i

~〈[H,Q]〉 , (22)

recovering our earlier result for the case that H is one of the pair of operators,and giving the general result for the time-dependence of the expectation valueof an operator. Unless H depends explicitly upon time, energy is conserved....

LLQM present yet further the striking argument that This result the aboverelationship between [H, f ] and (i/H)(Hf − fH) is also true for any twoquantities f and g ... (real/hermitian operators). The QM operator i (fg − gf)evolves to the classical ~ [f, g], which is duly tiny. They assert that This fol-lows at once from the fact that we can always formally imagine a systemwhose hamiltonian is g.!!

The presence of the i in the equivalence above, together with the other quan-tities real/hermitian, indicates that the commutator must be an imaginaryquantity. Consider the conjugate pair px, x , or any canonically conjugate

pair pi, qi. Since pi = −i~ ∂

∂qi,

i (piqi − qipi) Ψ = i (−i~)∂Ψ

∂qi= ~

∂Ψ

∂qi, (23)

which is duly real.

7 Uncertainty Relations

Following DQM, we demonstrate this constraint upon measurements of acanonically conjugate pair of observables:

Consider the commutator[K,F ] = iM , where K, F, and M are hermitianoperators, M and F a conjugate pair. In the most familiar context, M = ~,but Davidov's proof is more general; nb M need not be a constant.

33

We indicate the averages or expectation values of K, F, and M by 〈K〉, 〈F 〉,and 〈M〉 (eg 〈K〉 ≡

´Ψ†KΨ). Dening ∆K ≡ K−〈K〉 and ∆F ≡ F −〈F 〉 ,

we note that these obey the same commutation relationship as K and F:[∆K,4F ] = iM

We shall consider an auxiliary integral over τ (spanning the space of Ψ),depending upon an arbitrary parameter α,

I (α) =

ˆ|(α∆K − i∆F ) Ψ|2 dτ = 0. (24)

The integrand, a complex square, is non-negative, hence also the integral.Using the self-adjoint property of K and F, the integral is expanded to yield

I (α) =

ˆΨ† (α∆K + i∆F ) (α∆K − i∆F ) Ψdτ (25)

= α2⟨∆K2

⟩+ α 〈M〉+ 〈∆F 〉2 > 0 (26)

=⟨∆K2

⟩(α +

〈M〉2 〈∆K2〉

)2

+ 〈∆F 〉2 − 〈M〉2

4 〈∆K2〉> 0. (27)

We have completed the square on the 1st two terms; and note〈∆K2〉〈∆K2〉2

=

1

〈∆K2〉> 0, I(α) is non-negative; it is large for α large, either positive or

negative; and it has a minimum at α = − 〈M〉2 〈∆K2〉

, where the term [...]

vanishes. It follows that

〈∆F 〉2 − 〈M〉2

4 〈∆K2〉≥ 0,

〈∆F 〉2⟨∆K2

⟩≥ 〈M〉

2

4QED.

The most familiar case is M = ~, K = x, F = px, where

〈∆px〉2 〈∆x〉2 ≥~2

4. (28)

34

8 Consequences of the Uncertainty Principle

8.1 Phase Space

From the above we can recognize a hyper-space, conventionally called phasespace, pixellated as Harold has noted, into cells of volume element

∆x∆px∆y∆py∆z∆pz =~3

8. (29)

hat constant is really small! Fermi statistics allow only one fermion per cellof phase space (more on this under Dirac's 4-component electron). This isthe basis of the Thomas-Fermi model of atoms. Ultimately it explains whymaterial objects consume space and support themselves against pressure.Each electronic orbital requires its own pixel. Squashing an electron in ∆xraises ∆pxand hence the energy, which requires work.

With SR/Minkowsky in mind, one might add another factor of~2, for time

and energy. However, it is not clear in CM that t and H have the properstatus of a conjugate pair: Although they obey an H relation in QM, theydo not appear classically via a regular Lagrange transformation. In dis-cussion of such transformations with use of a generating function F (onwhich more below), GCM p.243 states a relationship which he notes, with-out demonstration, to be reminiscent of the relativistic result that iH/c isconjugate to x4 = ict. The issue arises repeatedly, in various contexts.

8.2 No velocity measurements!

[Following DWIQM (Ch. 8)]; As noted earlier, In non-relativistic QM, op-erators for velocity measurements do not occur! They would require nearbypairwise measurements of time and position. A rst measurement of posi-tion renders the momentum indenite, and hence the position thereafter. Wemust work with expectation values for a particle of mass m moving in onedimension, within potential V (x). Eects of the Uncertainty Principle areso pervasive as frequently to dictate the approach to calculations.

35

8.3 The relation between packet velocity and momen-tum

Preliminary: Begin with [Px, x] = −i~ (and trivial but soon signicant[x, Px] = +i~).Multiply [Px, x] = −i~ rst from the left, then from the right, by Px , toobtain P 2

xx − PxxP = −i~Px and PxxPx − xP 2x = −i~Px. Summing these

eqns., the cross-terms cancel: [P 2x , x] =P 2

xx− xP 2x = −2i~Px (An incidental

irrelevacy: This generalizes to [P nx , x] = −ni~P n−1

x ).

Now for the expectation values:

d 〈x〉dt

=d

dt

ˆψ†xψdτ =

ˆ (∂ψ†

∂txψ + ψ†x

∂ψ

∂t

)dτ, (30)

No explicit time dependence in x ; dτ ~ the function space. The time-

derivatives are∂ψ

∂t=

1

i~Hψ and

∂ψ†

∂t= -

1

i~ψ†H . Recall that H is self-

adjoint and that it operates to the left upon ψ†. Thus

d 〈x〉dt

=1

i~

ˆ ((ψ†H

)xψ − ψ†xH (ψ)

)dτ. (31)

Take H =P 2x

2m+V (x). Using the result for [P 2

x , x] above, and [H, x] = 0, we

have [H, V (x)] = 0, so that

d 〈x〉dt

=1

2m

1

i~

ˆψ†(−P 2

xx+ xP 2X

)ψdτ,

=1

2m

1

i~

ˆψ†[P 2x , x]ψdτ,

= − 1

2m

1

i~

ˆψ† (−2i~Px)ψdτ,

=〈px〉m

, (32)

giving the time derivative of the position expectation, the packet velocity,in terms of the momentum expectation.

36

8.4 Newton's 2nd Law of Motion

Referring to Preliminary above, begin instead with the commutator [x, Px] =+iH . It follows immediately that [x, P 2

x ] = +2iHPx.

In similar fashion to the above,d 〈px〉dt

is evaluated:

d 〈px〉dt

=d

dt

ˆψ†Pxψdτ

=

ˆ (∂ψ†

∂tPxψ + ψ†Px

∂ψ

∂t

)dτ

=1

i~

ˆ (−(ψ†H

)Pxψ + ψ†Px (Hψ)

)dτ

=1

i~

ˆψ† [Px, H]ψdτ.

The termP 2x

2min H commutes with Px, so the commutator includes only

V (x).

d 〈px〉dt

=1

i~

ˆψ† [Px, V (x)]ψdτ

=1

i~

ˆψ†[−i~ ∂

∂x, V (x)

]ψdτ

= −ˆψ†[∂

∂x, V (x)

]ψdτ

= −⟨∂

∂xV (x)

⟩. (33)

The rate of change of the momentum expectation is equal to the expectationof the force; this should be familiar.

8.5 The wave packet yielding theminimal uncertainty-product

The development follows the lines of DWICQM p. 129+, but as a preliminarywe convey a more general picture of Hilbert function theory:

37

Let there be functions f (τ) , g (τ),... upon a common parameter space τ. A concept of adjoint is dened; and a quasi-length product can bedened, isomorphic to a geometrical length product (a dot product). Webegin with a projected length product (units quasi-length squared), richlyanalogous to |F | cosϑ |G| for two vectors in ordinary space. It is expressed

as an integral over the space τ : f · g ≡ˆf (τ)† g (τ) dτ . The quasi-length2

of f is f · f =

ˆf †(τ)f(τ)dτ ; this is taken to be nite for each function.

Orthonormality is quickly appreciated, and an orthonormal basis set ri(τ);hence also the expansion of an arbitrary f(τ) onto such a function-set,wherein the integrals above play the role of the dot product.

Schwartz's Inequality asserts that very generally (f · g)2 6 (f · f) (g · g) .This is analogous to the assertion that cos θ ≤ 1 in |F |cosθ|G| above. Re-garding the expansion of f(τ) onto the full set ri(τ), we see that theprojection onto the single function g(τ) will typically miss a number of com-ponents, hence supporting the inequality. For such normalized functions, theequality is attained only if f(τ) is a unit complex number times g(τ) .

We consider now a single particle situated in 1-dimensional space, i.e. alongthe x -axis. The 'uncertainty' in the position of our particle must be givensome exact meaning. For the square of the uncertainty we adopt the devi-ation from the mean, expressed as ∆x ≡ x − 〈x〉. The expectation for thesquared deviation is⟨

∆x2⟩

=

ˆψ† (∆x)2 ψdτ =

ˆ(∆xψ)2 dτ. (34)

A similar expression for the squared deviation of the momentum px is⟨∆p2

x

⟩=

ˆ(∆pxψ)2 dτ. (35)

Then consider the Schwartz Inequality

⟨∆x2

⟩ ⟨∆p2

x

⟩≥∣∣∣∣ˆ ψ†∆x∆Pxdτ

∣∣∣∣2 = |〈∆x∆px〉|2 (36)

38

The operator within the expectation on the right is

∆x∆Px =1

2[∆x,∆Px] +

1

2(∆x∆Px + ∆Px∆x)

=i~2

+1

2(∆x∆Px + ∆Px∆x) (37)

Note that ∆x∆Pxhas one imaginary and one real (hermitian) term. Conse-quently the complex square of the expectation, |〈∆x∆px〉|2 , is just the sumof the square moduli of these two terms

|〈∆x∆px〉|2 =~2

4+

1

4〈∆x∆px + ∆px∆x〉2 . (38)

The term~2

4is inescapable via choices of the parameters, so 〈∆x2〉 〈∆p2

x〉 ≥~2

4. To attain the equality 〈∆x2〉 〈∆p2

x〉 =~2

4, the Schwartz Equality must be

achieved above, and1

4〈∆x∆px + ∆px∆x〉2 must vanish; DWIQM recognize

two distinct requirements here.

The 2ndcondition is modied, relative to the above discussion of normalizedfunctions, to f = αg , where α is some complex number. (DWIQM says anycomplex number [??]; well, surely not zero.)

This condition on f and g becomes ∆xψ = α∆Pxψ , to which the adjoint isψ†∆x = α†ψ†∆Px.

The condition for the Schwartz Equality |〈∆x∆px〉|2 = 0 may be written

as

ˆψ† (∆x∆Px + ∆Px∆x)ψdτ = 0, whereupon the proportionality givesˆ

ψ†(α†∆P 2

x + α∆P 2x

)ψdτ = 0 , or

(α† + α

) ˆψ†∆P 2

xψdτ = 0.

The integral must be positive denite (> 0); thus α must be pure imaginary.Any real part would give a result < 0.

Now it is possible to integrate ∆xψ = α∆Pxψ.

∆xψ = α∆Pxψ = α

(−i~ ∂

∂x− px

)ψ. (39)

39

The partials are unnecessary; and dx = d (∆x).

∆xψ = α

(−i~ dψ

d (∆x)− pxψ

)(40)

Multiply through by d(∆x)/ψ and transpose, obtaining

∆xd (∆x) + αpxd (∆x) = −i~αdψψ. (41)

A logarithm is recognized for ψ , thence an exponential. Note pxd (∆x) =pxdx. Thus

1

2d((∆x)2)+ αpxdx = −i~αd (lnψ) . (42)

Recalling that α may be an arbitrary imaginary #, choose α =i

~, consonant

with quasi plane-wave character.

1

2d((∆x)2)+ i

px~dx+ ln (C) = d (lnψ) , (43)

with C the constant of integration. Recall that ∆x ≡ x − 〈x〉; and thusdx = ∆x.

Note that neither the natural log nor the exp is allowed any net units withinthe argument.

Since the normalization condition is´ +∞−∞ ψ†ψdx = 1, any imaginary part

of C would be irrelevant (self-cancellingonly a shift of phase); nor will theimaginary part, the term in px , contribute to the integral. Thus we canproduce accord with the no net units principle by dividing the real term ind (lnψ) above by −2 〈∆x〉2 , without disturbing any normalization relativeto the 2nd term!. The minus is essential for convergence.

d ln (< (ψ)) = lnC − 1

4

d((∆x)2)〈∆x〉2

. (44)

Re-attaching the imaginary part

d lnψ = lnC −d((∆x)2)

4 〈∆x〉2+ i

px~

∆x. (45)

40

Then

ψ< (∆x) = Ce−d((∆x)2)

4 〈∆x〉2+ipx~

∆x

. (46)

For normalization purposes only, omit again temporarily the imaginary termin px

1 =

+∞ˆ

−∞

ψ†<ψ<dx = C2

+∞ˆ

−∞

e−

(∆x)2

4 〈∆x〉2 d∆x (47)

Note that < ∆x > is a constant.

1 = 2C2 〈∆x〉+∞ˆ

−∞

e−

∆x

2 〈∆x〉

2

d

(∆x

2 〈∆x〉

). (48)

Let u =∆x

2 〈∆x〉then 1 =2C2 〈∆x〉

´ +∞−∞ exp [−u2] du . The integral equals

√π (See @, below). Thus C2 = (2π 〈∆x〉)−1 , and C = (2π 〈∆x〉)−

12 .

Finally, our minimal packet is:

ψ = (2π 〈∆x〉)−12 e

−1

4

(∆x)2

〈∆x〉2+ipxx

~

. (49)

•Error in a standard text: DWIQM p.132 diers in the power of ∆x within

the expectation in the constant C : There C is displayed as: C =[2π < (∆x)2 >]−1/2

The CRC Concise Encyclopedia of Mathematics gives the Gaussian ProbabilityDistribution as

Pr (x) =1

σ√

2πe−

(x− u)2

2σ2 , (50)

which is equivalent to

Pr (x) =1√

2πσ2e−

(x− u)2

2σ2 (51)

41

and similar to DWIQM's expression.

The resolution of this issue lies in a subtle, even deceptive, and rather inter-esting distinction between the Gaussian Probability Distribution Pr and anexpectation value given as an integral over dx.

Pr is an ordinary probability distribution, normalized to 1 under summationor integration, here over

´d(instances) . The instances are spanned by dx

, with x running from −∞ to +∞; but here this mathematician's x inPr(x) is not the x of QM, nor does it bear units (length), or any particulargeometric meaning.

The integral for C2 above is dened with the dierential du of u =∆x

2 〈∆x〉,

running from −∞ to +∞.

In the normalization integral for C2, d∆x has geometric meaning, and henceunits = |length| . This unit and magnitude carry forth into the integral; butthis consideration is not reected in the

´du as dened.

Hence there is need for a factor in C2 of some natural measure of length.< ∆x > is such a measure, and therefore the constant C2 is to be awardedan extra factor ∆x . Thus C and ψ have been awarded an extra factor of√〈∆x〉 , relative to Pr.

It is gratifying (and conrmatory) that the same factor of −2 〈∆x〉2, appliedto < (ψ), delivers this leading factor

√〈∆x〉 in ψ and also brings accord with

the argument of the exponential such that the Gaussian form of the integralcan be recognized.

The error is assigned in that DWIQM, accepting the straighforward proba-bility identication with the Gaussian, have overlooked this last factor. Anexpectation value taken as an integral over a length x is not quite identicalto a probability distribution over some unitless parameter.

The integral

I ≡+∞ˆ

−∞

e−u2

du = 2

+∞ˆ

−0

e−u2

du ∝ C−2, (52)

because the integrand is symmetric about u = 0. The problem is addressed

42

with the double integral

1

4I2 =

+∞ˆ

−0

e−x2

dx

+∞ˆ

−∞

e−y2

dy (53)

(yes, ∝ C−4) , which is then taken to polar coordinates as

1

4I2 =

ˆ π/2

0

dθ

ˆ ∞0

e−r2

rdr =π

2

ˆ ∞0

e−r2

rdr =π

4

ˆ ∞0

e−r2

2rdr. (54)

With v ≡ r2,1

4I2 =

π

4

´∞0e−νdv =

π

4, so that I2 = π , and I =

√π . QED

9 Flow of Matter

9.1 Conserved Flow of Probability

(CTQM) We have identied ρ (q, t) ≡ ψ∗ (q, t)ψ (q, t) as the probability den-sity asserted by ψ at time t and at the position q in phase space; and we havenoted that

´ρ (q, t) dq = 1 . Though the integral is xed,ρ (q, t) may vary

locally, in time. But, because the integral is xed, we expect an equation of

continuity/conservation, in the nature of∂

∂tρ (q, t) +∇ · J (q, t) = 0.

Consider the Schrödinger equation i~∂

∂tψ (q, t) = − ~2

2m∇2ψ (q, t) + V (q, t),

where ∇2 ≡ ∇ · ∇. We discern immediately the divergence of a vector, thegradient

∂

∂tρ (q, t) =

(∂

∂tψ∗ (q, t)

)ψ (q, t) + ψ∗ (q, t)

(∂

∂tψ (q, t)

). (55)

The 2nd term is taken from ψ∗ times Schrödinger's equation, and the 1stterm from −ψ right-multiplied by the complex conjugate of Schrödinger'sequation; since V (q) is real, it cancels out

i~∂

∂tρ (q, t) = − ~2

2m

((∇2ψ∗ (q, t)

)ψ (q, t)− ψ∗ (q, t)

(∇2ψ (q, t)

))= − ~2

2m((∇ · ∇ψ∗ (q, t))ψ (q, t)− ψ∗ (q, t) (∇ · ∇ψ (q, t))) .

(56)

43

But we need ∇·, the divergence, operating upon an expression entire. Wehave, for any scalar a and any vector V the identities

∇ · (V a) = ∇ · V a+ V · ∇a (57)

∇ · aV = ∇a · V + a∇ · V (58)

Adding∇ψ∗ · ∇ψ −∇ψ · ∇ψ∗ = 0, (59)

we obtain

i~∂

∂tρ = ∇ ·

(− ~2

2m(∇ψ∗ψ − ψ∗∇ψ)

), (60)

∂

∂tρ = ∇ ·

(− ~

2im(∇ψ∗ψ − ψ∗∇ψ)

), (61)

so that the conserved current is

J =~

2im(ψ∗∇ψ −∇ψ∗ψ) , (62)

where in the 2nd term we have commuted the two factors.

For any function ψ oered as a wave equation, J must be positive de-nite. In the development of Dirac's model for the relativistic free particleby BDRQM, this becomes a critical issue. Meaning? My understanding isthat he means essentially that the probability must share the direction of themomentum.

9.2 A ux of free particles encounters a potential wall

This poses a scattering problem, appropriate for unit-plane-wave normaliza-tion (see above). We are given a potential V (x) = 0 for −∞ < x < 0 and aconstant valueV (x) = V > E where E is the particle energy, for 0 < x < +∞. From the left there approaches an incident ux ψ(x, t) = expi(ωt −kx) of particles of energy E, obeying Schrödinger's equation i~

∂

∂tψ (q, t) =

− ~2

2m∇2ψ (q, t) + V (q, t)ψ (q, t). Taking ψ (x, t) = eiωtu (x) , this reduces to

∇2u (x) +2m

~2(E − V (x))u (x) = 0, (63)

44

with eiωt cancelled throughout.

We distinguish two cases, characterized each by a real positive constant, kor κ:

1. For E − V (x) > 0, we can writed2u

dx2− k2 (x)u = 0, with k (x) =

1

~(2m (E − V (x)))

1/2, so that k2 > 0. This has solutions u (x) = e±ikx

, where the minus sign is suited to describe our initial ux approachingfrom the left.

2. For V−E > 0, we can writed2u

dx2+κ2 (x)u = 0, with κ (x) =

1

~(2m (V − E))

1/2,

so that κ2 > 0 also. This has solutions u (x) = e±κx , where again theminus sign is suited to describe ux proceeding to the right.

The discontinuity of V(x ) at x = 0 bears more detailed discussion: Considera series of models in which the discontinuity is softer, but with the seriesapproaching the discontinuous case, the steep limit. For each such model,pursue the following:

For x approaching 0 from negative values, case (1) applies: As V(x) increasesfrom 0, V(x) passes through the value E (recall that the constant V>E ),

where k → 0 and the wavelength λ =2π

kbecomes indenitely large; so the

slope of u(x) becomes indenitely small at x = 0 , regardless of the initialphase. In the steep limit, the plunge ofλ, hence of the slope of u(x), occursaltogether abruptly. Thus the initial plane wave uI(x) = e±ikx continues tox = 0 (subscript I denotes incident).

For x increasing above 0, case (2) applies: V (x)−E is positive and increasing;

in the steep limit, κ attains its limiting value1

~(2m(V (x)− E))

1/2 abruptly.

The right-hand solution u(x) = e±κx begins abruptly at x = 0 and continutesto x = +∞.

We seek a limiting net solution including the incident wave of amplitude 1, areected wave of amplitude R travelling backward (within the space x < 0),and a transmitted amplitude T continuing forward (within the space x > 0); these amplitudes pertain at x+ 0.

Altogether, then, we have three waves:

45

1. The incident wave approaching, for x < 0 ,uI = e−ikx.

2. The reected wave, also for x < 0 , uR = −Reikx.

3. And a transmitted wave for x > 0 , uT = (1−R) e−κx.

Note for 2) the phase shift of π , as R → −R , familiar from optics. Ifthe magnitude of R is unity, we have perfect reection; T=0, and the netu|x=0−ε = 0.

9.3 Hamilton's wave equation; the classical eikonal:

Further toward our goal to convey the relationship between QM and CM,we drive for an intermediate objective, long-since promised: Via Hamilton'sPrincipal Function S and his Characteristic Function W with the recognizedlimitation∂H/∂t = 0 (ie a fully isolated system) we show that all coordinatesare cyclical. Thereupon we discover a scalar wave equation evolved from acontext of QM.

The power of Legandre transformations and classical variational principles isfurther to be exploited. Here we hop along a series of points from GCM, butwith long stretches of abridgement.

NB in general: lower case ⇒ before transformation, upper case ⇒ after.At some points hereafter the single symbol qi is used to represent the setqi, and pi for the set pi; but these points should be obvious.

The transformations discussed heretofore have the form qi → Qi and pi → Pi;these are termed point transformations (they don't scramble q's and p's).But presently others, more general canonical or contact transformations,interest us. It is required that there be some energy function K (Qi, Pi,t)

(analogous to H ), such that Qi =∂K

∂Piand Pi = − ∂K

∂Qi

. The heavy dots

represent time-derivatives. Our variables must respect the variations

δ

ˆ t2

t1

(∑piqi −H (qi, pi, t)

)dt = 0, (64)

δ

ˆ t2

t1

∑i

(PiQi −K (Qi, Pi, t)

)dt = 0. (65)

46

The two integrands need not be equal; rather they dier at most by a totaltime derivative of some arbitrary generating function F. We employ one ofHamilton's four functions F1,2,3,4. These contain in toto 4n+1 quantities: the2n original coordinates and momenta qi and pi; the 2n transformed quantitiesQi and Pi ; plus t . Only 2n of the 4n are taken as the independent variables,the others being taken as consequential. GCM discusses the four functionsF1,2,3,4, diering in those choices, as F1 (qi, Qi, t), F2 (qi, Pi, t), F3 (pi, Qi, t) ,and F4 (pi, Pi, t). Of theseF2 (qi, Pi, t) concerns us now. It is obtained by adouble Lagrange transformation; now treating the integrand for the variation∑

i

(piqi −H (qi, pi, t)) =

∑(PiQi −K (Qi, Pi, t) +

d

dt(F2 (qi, Pi, t)−

∑QiPi)

).(66)

Expanding the nald

dt

∑, we note cancellation/replacement a' la Lagrange;

and withdF2

dtdeveloped, we obtain∑

i

(piqi −H (qi, pi, t)) =

−∑

i

(QiPi −K (Qi, Pi, t) +

∂F2

∂qiqi +

∂F2

∂PiPi

)+∂F2

∂t. (67)

Equating the coecients of our dotted independent variables qi, and Pi, weobtain the transformation equations:

pi =∂F2

∂qi, Qi =

∂F2

∂Pi, with the transformed energy K = H +

∂F2

∂t.

We now set up and justify the hamiltonian-Jacobi equation for Hamilton'sPrincipal function S, a case ofF2 (qi, Pi, t). We set the transformed hamilto-nian K ≡ 0, which ensures that the new variables are constant in time:0 =∂K

∂Pi= Qi and 0 = − ∂K

∂Qi

= Pi.

47

Also 0 = K = H +∂F2

∂t, so that 0 = H (qi, pi, t) +

∂F2

∂t. From the lines just

above, this may be written as the hamiltonian-Jacobi equation

H

(qi,∂F2

∂qi, t

)+∂F2

∂t= 0, (68)

with the i's now representing two strings (as anticipated above), for both theq' s and for the p's.

This is a partial dierential equation in n + 1 variables, the qi and t .There must be (justication!) a solution for F2 (qi, Pi, t), which we name asHamilton's Principal Function S (qi, Pi, t) , of which GCM writes Of course(this) only provides the dependence upon the old coordinates and time; itwould not appear to tell how the new momenta are contained in S . Indeedthe new momenta have not been specied, except that they must be con-stants. However, the nature of the solution indicates how the new Pi are tobe selected.

The solution must have n + 1 constants, α1, α2, ...αn+1. However, the H-Jequation contains only derivatives of S, not S itself; hence one of the con-stants, say αn+1, is just an additive one, irrelevant to the solution, and maybe omitted.

Hence a complete solution for F2 can be written as S (qi, α1, α2, ...αn, t).We are at liberty to take these constants to be the new (constant) mo-menta: Pi = αi . This does not interfere with their connection with theinitial values of the qi and pi at time zero.

Congruent with our earlier essential dismissal of time dependence in H (atleast for the initial treatment of typical QM problems), we presently take∂H

∂t= 0; in this the results are less than general. The H-J equation be-

comes, with S (qi, α1, α2, ...αn, t) the solution for F2: H

(qi,

∂S

∂qi

)+∂S

∂tt = 0

. Immediately upon taking∂H

∂t= 0, we have achieved separation of vari-

ables! That is, the time variable can be separated by assuming a solution forS of the form

S (qi, α1, α2, ...αn, t)) = W (αi, qi)− α1t , whereupon we nd H

(qi,∂W

∂qi

)=

α1. Limiting (GCM p. 307) to cases H = constant = total energy , we haveS (qi, Pi, t) = W (qi, Pi)− Et.

48

For the central geometrical-optical ray, then, (and for its neighborhood inthe sense of the δ-variation) and in this special generalized coordinate setwe have a wavefront in conguration space, with the phase S proceedingaccording to −α1t . We may anticipate a wave equation arising.

Since W is independent of time, surfaces of constant W in congurationspace have xed locations in this space.

A surface of constant S must at a given time coincide with some surface ofconstant W . However, the correspondence changes with time according tothe equation just above for S (qi, Pi, t).

A surface of constant S surface moves at velocity u =dS

dt(constant S⇒locally

constant over the surface,;constant in time).

In the same time dt that the S -surface travels distance udt , it travels to anew W -surface W + dW , with dW = Edt . The change dW = |∇W | dS,

also, so that u =dS

dt=

dW/ |∇W |dW/E

=E

|∇W |. The Pi are just a stack of

constants α.

For simplicity we consider just the motion of a single particle three dimen-sions, moving in potential V (qi).

The gradient magnitude |∇W | is obtained from

3∑i=1

(p2i + V (qi)

)=

3∑i=1

(1

2m

(∂W

∂qi

)2

+ V (qi)

)= E (69)

as (∇W )2 = 2m (E − V (qi)), the Hamilton-Jacobi Equation for this singleparticle.

And nally u =E

|∇W |=

E√2m (E − V )

=E

p=

E

mv. The velocity of

a point on a surface of constant phase S is inversely proportional to theparticle velocity. This may be familiar from the QM behavior of packets inregard to phase and group velocities.

For cases more complex than that of a single particle, a parallel developmentcan be indicated. GCM (pp 228,310) opens this issue via the Least ActionPrinciple, with a ∆-variation of

´ t2t1

∑i piqidt (not developed here in full).

This leads via some dierential geometry to a more general expression for

49

dt in terms of an arc length, as dt =dρ√2T

and thence (GCM p. 310) to

S -surface velocity u =E√2T

.

The scalar wave equation of optics,

∇2φ− n2

c2

∂2φ

∂t2= 0, (70)

(n the index of refraction, c the velocity of light), is satised by a planewave solution ϕ = φ0e

i(k·r−ωt). The wave number k and the frequency obey

k =2π

λ=

nω

c. We adopt the axis z; k0for the wave number in vacuum.

Adopting the z -axis, we may write ϕ = ϕ0eik0(nz−ct).

The index n is taken to vary slowly in space, accordingly distorting andbending the wave. Seeking a solution as near to a plane wave as possible, weadopt

φ = eA(r)+ik0L(r)−ct, (71)

where L is an eective optical path length, called the eikonal. GCM isabout to win Planck's constant out of this!

GCM then evaluates the Laplacian ∇2φ ; this has several terms but is man-ageable. The lhs of the wave equation has real and imaginary parts, both ofwhich must vanish. Given that n varies only slowly, the prominent termis that term in the imaginary part which does not contain A . The result is(∇L)2 = n2, the eikonal equation of geometrical optics. The similarity tothe Hamilton-Jacobi Equation does not imply that L and W are equivalent,[only] that they are proportional to one another.

(Accordingly), S = W − Et must be proportional to the total phase of the

light wave described by equation above: k0 (L− ct) = 2π

(L

λ0

− νt)

Hence the particle energy E and the wave frequency ν must be propor-tional;... we denote the constant ratio by the symbol h, obtaining E = hν. The wavelength and the frequency are connected by λν = u . Thus, by

u =E√2mT

obtained above, we have λ =u

ν=E/p

E/h=h

p

Next GCM produces the Schrödinger Equation!

50

The scalar wave equation, in an other-than-optical context, may be writtenas

∇2φ− 1

u2

∂2φ

∂t2= 0. (72)

If the time-dependence is taken as e−iωt , we obtain the time-independentwave equation

∇2φ+4π2

λ2φ = 0. (73)

Corresponding to the wave amplitude in optics there will be some quantityψ in the wave theory of mechanics which must satisfy an equation of the

same form... . But now λ =h

p=

h√2m (E − V )

, so that we have the result

∇2ψ +8π2m

h2(E − V )ψ = 0, (74)

the time-independent Schrödinger equation. For subsequent ready recogni-tion, we rewrite this as

~2

2m∇2ψ + E − V = 0, (75)

or as

i~∂ψ

∂t= − ~2

2m∇2ψ + V (x). (76)

With the eikonal ray for a material particle now appreciated, recognize thatit follows the behavior of the ideal ray discussed above in connection withthe fundamental mode of a confocal laser.

9.4 Hamilton's place in history:

GCM states (p.314) that The equivalence of the Hamilton-Jacobi and eikonalequations was rst realized by Hamilton in 1834.

GCM notes here that some have suggested that Hamilton almost discoveredQM, had he reected whether the Poisson Bracket was truly zero (see zerovs. very small, far above). Then he argues, to the contrary, that Hamilton

51

lacked the authority [of experimental evidence, one infers]. Authorityscarcely seems needed in the culture of contemporary physics, to advancesuch a conjectural theory in such a context.

Likely more to the point: There was no concept abroad, in Hamilton's epoch,of De Broglie's matter-waves. [De Broglie himself only mentioned his ideabriey, in a footnote to his thesis; he didn't amplify upon the implications.Finally, per Google/Davisson-Germer, the initial interest of DG was solelyin the surface of nickel.]

Cf the remark in the Introduction that a ...chancy arrival at a particularlyproductive perspective deserves emphasis..., as well as the innate gifts ofworkers.

9.5 Quantummechanical development of the eikonal viathe WKB approximation:

Hamilton developed the Characteristic Function W and the Principal Func-tion S, thence the eikonal, from a classical perspective, developed above atlength. Capturing the wave aspect of propagation of a material object, itanticipates QM, aording a furthest reach of CM. From an opposite perspec-tive, the Characteristic Function W can be produced via a sort of degradationof QM, the WKB approximation; this follows*.

Let the functional dependences of the wave function ψ be expressed within its

logarithm [SQM, p.184]: That is, in the Schrödinger equation E = i~∂ψ

∂t=

− ~2

2m∇2ψ + V (r))ψ , write ψ as Aei/~W (r,t).

Dierentiating once with respect to t ; and, separately, twice with respect tor, one obtains

∂W

∂t+

1

2m(∇W )2 + V (r)− i~

2m∇2W = 0. (77)

Here the common factor Aei/~W (r,t) has been cancelled throughout.

In the classical limit, ~ = 0, and this equation is the same as Hamilton's

equation (earlier) for the Principal Function W,∂W

∂t+ H (r, p) = 0 , where

p = ∇W .

52

Taking ψ to be an energy eigenfunction, ψ = u (r) e−iEt/~, W can be writtenW (r, t) = S (r)−Et; so we have recovered Hamilton's Action, as well as hisPrincipal Function. And u (r) = AiS(r)/~. We recognize the kinetic energy,the total energy, and the potential energy in the successive terms of [SQMeqn 28.2, p.185]

1

2µ(∇S)2 − (E − V (r))− i~

2µ∇2S = 0. (78)

The WKB method obtains the rst two terms (one term beyond the classicalexpression) of an expansion of S in powers of ~ , in the one-dimensional case(to which we now collapse).

With u (r) = AiS(x)/~, Schrödinger's equation reduces to one of two forms

d2u

dx2+ k2 (x)u = 0, k2 > 0, (79)

d2u

dx2+ κ2 (x)u = 0, κ2 > 0, (80)

so that k and κ are always real, putting k(x) = +1/H[2µ(E − V (x))]1/2, orκ(x) = 1/H[2µ(V (x)− E)]1/2 in the respective cases.

We focus upon the case k > 0 , returning to the logarithm of u(x) [ie, againcancelling the factor AiS(x)/~ which appears in each term; such constants arerepeatedly absorbed...] and using primes for dierentiation

i~S ′′ − S ′2 + ~2k2 = 0. (81)

We apply an expansion of S to rst-order in ~: S = S0 + ~S1to get

i~ (S ′′0 + ~S ′′)− (S ′0 + ~S ′1)2

+ H2k2 = 0. (82)

Then writing an equation for each power of ~

− (S ′0)2

+ 2µ (E − V (x)) = 0, ~0 term,

iS ′′0 − 2S ′0S′1 = 0, ~1 term,

plus terms of yet higher powers of ~.

53

The rst, restated as S ′0 = ~k, integrates toS0 (x) = ±~´ x

0k (y) dy , where

the lower limit, a constant, is absorbed within A.The second is rewritten as

S ′1(x) =i

2

S ′′0 (x)

S ′0 (x)

=i

2

1

S ′0

dS ′0dx

=i

2

d (lnS0)

dx(83)

S1(x) =i

2[lnS0 (y)]x0

=i

2ln

~xˆ

0

k (y) dy

(84)

= i/2d/dx(1/HdS ′0(x)/dx)/1/HS ′0(x) = i/2d/dx[ln(k(x)−lnk(0)] = i/2d/dx[ln(k(x)], again ignoring the constant. Thus S1(x) =

´ y=x

y=0dS ′1(y) =

´ y=x

y=0d[lnS ′0(y)] =

i/2[lnk(x)− ln(k(0)] = i/2[lnk(x)] , yet again ignoring the constant.

Schi writes We thus obtain to this order of approximationu(x) = Ak−1/2exp(±i

´ xkdx for V < E and, u(x) = Aκ−1/2exp(±

´ xkdx for

V>E.

It must not be supposed that H2 is so small that this approximation is suitablefor all practical purposes. Schi points out the following: Since S0 is amonotonic increasing function so long as k does not vanish, the ratio HS1/S0

is small if HS ′1/S′0 is small. Thus we expect [the method] to be useful in that

part of the domain of x where |HS ′1/S ′0| <<1.

Now the DeBrolie wavelength λ is 2π/λ , so that [the inequality] can bewritten λ/4π|dk/dx| , which means that the fractional change in k (or in thewavelength) in the distance λ/4π is small compared to unity.

It is apparent that [the inequality condition] is violated near the turningpoint of the classical motion, where V (x) = E, k and κ are zero, and thewavelength is innite.

54

10 Angular Momentum

10.1 Types of Angular Momentum

An electron circulating about a nucleus exhibits orbital angular momentuml.

Fundamental unit particles have another sort of angular momentum, intrin-sic spin, usually termed simply the spin s. This is of a sort quite dierentfrom l and will require extensive discussion (below). The totals for severalelectrons are S and L. The total electronic angular momentum of an atom isJ. The above and other types to follow are taken as measures applied to theunit H of angular momentum (often taken implicitly, unstated).

For nuclei, aggregates of protons and neutrons (today themselves consideredas assemblies of three quarks), the net intrinsic spin is denoted by I; Thetotal angular momentum of an atom, including nuclear spin and hypernestructure, is F.

A rigid molecule may rotate about an axis of symmetry, with orbital angularmomentum J. A methyl group CH3 may twist about the axial bond con-necting it to the rest of its parent molecule, either in hindered rotation orin a rocking motion; these both are classied as vibrations (!), though inthe former case they carry angular momentum. Molecular spectroscopy en-tails several intermediate variables: the electronic angular momentum abouta linear axis Λ , the angular momentum P about an axis of symmetry, etc(texts of G. Herzberg).

10.2 Quantization with respect to Rotation

The operator for linear momentum along axis x is~i

∂

∂x, and a representative

eigenfunction appears as the spatial factor φ0ei(k·r) in the familiar propagat-

ing plane wave ϕ = φ0ei(k·r−ωt).

The operator for angular momentum is~i

∂

∂θ, and the corresponding eigen-

function is ϕ = φ0eijθ. But the analogy between x and θ is not complete.

Consider special relativity: Not only is there no specially directed axis, nor aspecial position along an axis, there are more particularly no ducial marks

55

as are found upon a ruler no special value for the dierence between twopositions.

With regard to rotation, however, there is in SP , in contrast, a special irrota-tional frame (consider centrifugal force). And for rotation about an axis, pas-sage through 2π is equivalent, in regard to all observable features/quantities,equivalent to no rotation at all; certainly this constitutes a special dierence

between two positions. The j-value1

2, given θ = 2π , yields a phase factor of

φ0eijθ = φ0e

i1/22π = φ0eiπ = −φ0. Thus −1 is the rst real eigenvalue as j is

increased from zero. The norm of the j =1

2state isφ†φ ; this includes factors

(−1)(−1) = +1, ie unity, just as for θ = 0. And for any hermitian operatorΩ , the expectation value φ†Ωφ is again unchanged. Thus -1 is an accept-

able eigenvalue for the rotation operator, and j =1

2conveys an acceptable

eigenstate. Any half-integral or integral j is acceptable.

10.3 Experimental Discovery and Theoretical Accountfor Intrinsic Spin

Intrinsic spin was discovered experimentally (vs. predicted) perhaps origi-nally via Zeeman's eect (1896): Spectroscopic lines were split by a magneticeld. Particularly notable also is the Stern-Gerlach experiment (1922): Sil-ver atoms were deected in an inhomogeneous magnetic eld, those with

Sz =1

2~ in one direction and those with Sz = −1

2~ oppositely.

The following is from Google, on intrinsic spin: According to the prevailing

belief, the spin of the electron or some other particle is a mysterious

internal angular momentum for which no concrete physical picture

is available, and for which there is no classical analog. However,

on the basis of an old calculation by Belinfante [Physica 6 887 (1939)],

it can be shown that the spin may be regarded as an angular momentum

generated by a circulating flow of energy in the wave field of the

electron. Likewise, the magnetic moment may be regarded as generated

by a circulating flow of charge in the wave field. This provides

an intuitively appealing picture and establishes that neither the

spin nor the magnetic moment are internal they are not associated

with the internal structure of the electron, but rather with the

56

structure of the field. Furthermore, a comparison between calculations

of angular momentum in the Dirac and electromagnetic fields shows

that the spin of the electrons is entirely analogous to the angular

momentum carried by a classical circularly polarized wave. Mod-ern experiments with very high-energy electron beams suggest no limitationon the localization of the electron (hence vanishing size), given sucientmomentum.

10.4 Symmetry upon Exchange of Particles; Fermi andBose Statistics

Half-integral intrinsic spin is associated with Fermi statistics, and integralvalues to Bose statistics: For fundamental particles the Fermi type, theexchange of any two identical particles exhibits antisymmetry, a change ofsign in the net wavefunction. For Bose entities, the sign remains the sameunder such exchange. For two particles, eg, using subscripts for the functionsand superscripts for the particle coordinates, we have Ψ1

aΨ2b = ±Ψ2

aΨ1b ,

Bose/Fermi; multiple identical Fermions entail determinant forms. Thoughone describes a Fermion as having spin s , quantitatively the value is Hs .In the limit of classical mechanics this vanishes as ~→ 0.

Certainly the proposition that Nature submits to this limitation is supportedby all pertinent experiments, these representing an enormous weight of re-sults. Pauli claimed to have proved the proposition via SP, and stoutlydefended his analysis through the rest of his life, against all opposition. Inthe subsequent decades, the matter has been studied in great depth. I. Duckand E. C. G. Sudarshan have written a review at book-length, expresslyclaiming for themselves no new results (Pauli and the Statistics Theorem,World Scientic, NJ, 1997; 503 pp.). In Preliminary Remarks on p. 3, theywrite Everyone knows the Spin-Statistics Theorem, but no one understandsit, eg Feynman. On p. 4 one nds The Pauli result does not explain thespin-statistics relation and cannot.

57

10.5 Orbital Angular Momentum; Summation over Con-tributions to Angular Momentum

Vectors describing orbital angular momentum, a term used for the moreclassical sort r × p take on only positive integral values. They follow thestructure of the Spherical Legendre Harmonics.

There are fairly simple rules (see further below) for the addition of distinctcontributions (dierent degrees of freedom) to orbital angular momentum L,absent spin S; and there are entirely parallel rules for addition of spins S, ab-sent L. When both are present, construction of the total angular momentumis more complicated. Depending upon the strength of spin-orbit coupling,the models range:

1. from l,s coupling (Russel-Saunders spins s accumulated and l's ac-cumulated rst, respectively to S and L, then S + L = J );

2. to j,j coupling (s and l of each electron summed to j , then the j'ssummed to J); and

3. with a continuum of intermediate cases (low-lying states of Ne, forinstance, require such treatment).

Intrinsic spin ~S , with only modest values of S (excluding magnetic solids),vanishes in the classical limit ~ → 0 . But orbital angular momentum, ofthe nature r × p , may in principle take arbitrarily large integral multiplesof H. Planck's constant enters only in the commutation relation [Lx, Ly] =i~Lz (see below), and in two similar relations with cyclic permutation of theindices. For large values, this occasions only a relatively small fractionaluncertainty product.

10.6 Commuting Operators for Angular Momentum

An angular momentum vectorJ (either integral or half-integral) correspond-ing to the the scalar index j is limited to just two simultaneously diagonizableoperators, one for the square of the vector, J2 = J2

x + J2y + J2

z , and one for asingle one of the three axial components, conventionally Jz ; the three scalar

58

components do not commute pairwise with one another. To restate: The si-multaneously diagonalizable variables belonging to a vector J correspondingto index j are limited to J2 and Jz , with the pair written as J2, Jz ; Jxand Jy are not welcome in this company. While Jy and Jz are indeed ob-servables, they are excluded from the simultaneous-eigenfunction pair. Thescalar index j, while it enumerates the eigenvalues, is not itself even an ob-servable. Jz is indeed one of the eigenvalue pair, and it further serves as anindex on 2j + 1 substates. Whether the net J belongs to a single particle orto an assemblage; whether of spin-only (S), orbital-only (L), or mixed type;whether J is integral or half-integral Jz takes on values from −J to +J ,with uniform spacings of one unit.

Demonstration of the above (from CS p. 43, eqn 6): Among several vectoridentities involving commutators is found: [A ·B,C] = A · [B,C]− [C,A] ·B .To verify this, retreat to component coordinate indices, as AiBiCj−CjAiBi =?Ai[BiCj −CjBi]− [CjAi−AiCj]Bi = AiBiCj −AiCjBi−CjAiBi +AiCjBi

. The 2nd and 4th terms cancel, QED.

Now let both A and B =J ; and let C = Jz . Since J · J = J2 , we have[J2, Jz] = [J · J, Jz] = (J · J) Jz − Jz (J · J) = J2 (Jz − Jz) = 0 , QED; asclaimed, J2 commutes with any single one of the coordinates of J . Thepartner to J2 is conventionally taken as Jz.

10.7 Corollaries of 10.6

10.7.1 Commutators connecting the Cartesian coordinates of theangular momentum

As noted above, the scalar components of an angular momentum vectorJ do not commute pairwise. In fact we have the commutation relations[Jx, Jy] = i~Jz ; [Jy, Jz] = i~Jx; and[Jz, Jx] = i~Jh. This generalizes theresult of the preceding section for L.

59

10.7.2 Raising and lowering Jz to complete the set of eigenfunc-tions

Consider the operators Jx± iJy , applying Jz to either of them from the lefton both sides of the net equality

Jz (Jx ± iJy) = JzJx ± iJzJy= JxJz − i~Jy ± i (JyJz − i~Jx)= (Jx ± iJy) (Jz + ~) , (85)

[Jz, (Jx ± iJy)] [Jz, (Jx ± iJy)] = (Jx ± iJy) ~. (86)

Using case to distinguish the operators J2 and Jz from their eigenvalues j2

and jz (and using γ for any other eigenvalues), we exercise the operators oneach side of the prior equation upon a general wavefunction Ω (γ, j2, jz)

Jz (Jx ± iJy) Ω(γ, j2, jz

)= (Jx ± iJy) (Jz + ~) Ω

(γ, j2, jz

),

Jz((Jx ± iJy) Ω

(γ, j2, jz

))= (Jx ± iJy)

((Jz + H) Ω

(γ, j2, jz

)),

=((Jx ± iJy) Ω

(γ, j2, jz

))(jz + ~) . (87)

Writing Ω± (γ, j2, jz) for (Jx ± iJy) Ω (γ, j2, jz) , and noting that the eigen-value and ~ , constants, commute with everything, we nd

The notation suggests that the operators (Jx ± iJy) may be recognized asoperators promoting/demoting jz by one unit ~. This is true unless theresult Ω± (γ, j2, jz) is zero. Zeroes occur when the + operator would pro-mote an initial state Ω (j2, jz = j) , or when it would demote an initial stateΩ (j2, jz = −j). This is achieved because each of Jx and Jy yield zero for apure state of jz = ±j ; the result is quite satisfactory; there exist no statesjz > +j nor jz < −j .The construction does not yield normalized eigenstates; normalization mustbe pursued subsequently, as Ω± (γ, j2, jz)

†Ω± (γ, j2, jz) = 1.

10.7.3 Summing multiple degrees of freedom in angular momen-tum

Suppose now that we wish to map out the states belonging to the physi-cal summation of two j-values, j1 and j2, representing dierent degrees of

60

freedom (arbitrarily we take j1 ≥ j2 , for specicity). For the substates ofextremal jz (max or min), the solutions are trivial product states.

The state of maximal jz is

Ω((j1 + j2)2 , jz = j1 + j2

)= Ω1

(j2

1 , j1z = j1

)()Ω2

(j2

2 , j2z = j2

). (88)

And the state of minimal jz is

Ωdiff

((j1 − j2)2 , jz = j1 − j2

)= Ω1

(j2

1,j1z = j1

)Ω2

(j2

2 , j2z = −j2), (89)

recalling that j1 ≥ j2. The expressions suce also for j2 = j1, yieldingrespectively jz values of 2jz and 0.

Beginning with one of these extremal states, the other jz substates may begenerated stepwise, according to section (2) just above.

Altogether, we are thus armed, in principle, to synthesize states and substatesfor any simple nite summation over orbital and spin functions (but seeremarks above re j,j coupling).

11 The Dirac Relativistic Theory of the Free

Particle (Fermi statistics)

(following BDRQM, rather than Dirac)

11.1 Opening sally at the problem

Intrinsic spin arises in intimate association with special relativity. We shallnd that relativistic theory implies spin, whether or not high velocities areevident.

For an free particle, the nonrelativistic hamiltonian is H = p2/2m .

The transition to quantum mechanics is achieved with the transcriptionHiH∂/∂t , and pµ(H/i)∂/∂xµ.

This transcription is Lorentz covariant, since it is a correspondence betweentwo contravariant four-vectors pµiH∂/∂xµ .

61

This leads to the nonrelativistic Schrödinger equation, NRSE,iH∂ψ(q, t)/∂t =−H2/2m∂2/∂2xµψ(q, t)

Like position x, y, z , momentum is an observable. Hence ψ may be afunction of this variable, but a relativistic theory calls for a four-momentum.From BDRQM: According to the special theory of relativity, the total energyand momenta transform as components of a contravariant four-vector pµ =(p0, p1, p2, p3) = (E/c, px, py, pz).

The length is∑

3µ=0pµp

µ = E2/c2 − pp = mc2 ; this is the Klein-Gordonequation. Bold now with the dot indicates three-vector momentum. Thelength of the four-vector equals the invariant rest-energy mc2.

Following this, it is natural to take as the hamiltonian of a relativistic freeparticle H =

√p2c2 +m2c4 , and to write for a relativistic quantum analogue

to the NRSE above the candidate RSE iH∂ψ/∂t =√−H2c2∇2ψ(q, t) +m2c4ψ.

Unfortunately this is unsatisfactory; it treats time and space asymmetrically,and If we expand it, we obtain an equation containing all powers of thederivative operator and thereby a nonlocal theory.

We undertake to justify squaring the operators an each side of our candidateRSE (DQM does not include the following justication). BDRQM observesthat our candidate has the formAψ = Bψ , and equivalently Bψ = Aψ . Left-multiply the 1st equation byA and the 2nd by B, to obtain respectivelyAAψ = ABψ and BBψ = BAψ , then subtract. Provided that A and Bcommute, as indeed they do here, we have indeed justied A2ψ = B2ψ as avalid equation.

We have thus obtained −H2∂2ψ/∂t2 = (−H2c2∇2 + m2c4)ψ. Now from

BDRQM: This is recognized as the classical wave equation ψ+(mc

H)2 = 0

, where ≡ ∂/∂xµ∂/∂xµ . [∇ represents the gradient in 3 dimensions, and

∇2 the Laplacian; in a break with consistency, per BDRQM, representsthe 4-dimensional Laplacian, of 2nd order.]

As it develops, the multiplication to accomplish squaring yields a negative-energy solution, spurious relative to our initial intent; but that evolves intothe divine gift of the antiparticle.

Our rst task is to construct a conserved current, since [our equation in

62

] is a second-order wave equation and is altered from the Schrödinger form...upon which the probability interpretation in the nonrelativistic theory isbased. This we do in analogy with the Schrödinger equation, rst left-

multiplying[ +(mc/H)

2]ψ = 0 byψ∗, then left-multiplying the complex

conjugate,[ +(mc/H)2

]by ψ ; and nally subtracting, to obtain: ψ∗ ψ−

ψ ψ∗ = 0 , or ∇µ(ψ ∗ ∇µψ − ψ∇µψ∗) = 0, or

∂/∂t

[iH

2mc2(ψ ∗ ∂ψ/∂t− ψ∂ψ ∗ /∂t)

]+ div

H

2im(ψ ∗ gradψ − ψgradψ∗) = 0

. [Equation EQIT, equation quadratic in time]

We would like to interpretiH

2mc2(ψ ∗ ∂ψ/∂t− ψ∂ψ ∗ /∂t) as a probability

density ρ . However, this is impossible, since it is not a positive deniteexpression. For this reason we follow the path of history [refs. Schrödinger,Gordon, Klein, Dirac] and temporarily discard [Equation EQIT] in the hopeof nding an equation in rst order of the time derivative which admits astraight-forward probability interpretation as in the Schrödinger case. Weshall return to Equation EQIT above, however.

Although we shall nd a rst-order equation, it still proves impossible toretain a positive denite probability density for a single particle while at thesame time providing a physical interpretation of the negative energy root ofthe equation above.

Therefore [our Equation in the Square , EQIS, or equivalently EQIT] ,also referred to as the Klein-Gordon equation, remains an equally strongcandidate for a relativistic quantum mechanics as the one which we nowdiscuss.

11.2 The Dirac equation

Continuing quotes from BDRQM: We follow the historic path taken by Diracin seeking a relativistically covariant equation of the form of [the NRSEabove] with positive denite probability density. Since such an equation islinear in the time derivative, it is natural [or so it was to Dirac] to attempt toform a hamiltonian linear in the space derivatives as well. Such an equationmight assume the form:

63

The genius in this approach lies in the dierence between the treatment of thetime/energy/mass term and that of the momenta: The former is just once(ie linearly) introduced, in the hamiltonian operator, whereas the momenta,resident in the ψ∗ and ψ, attack twice, both from the left and the right;each item is linear, but the result is quadratic for the momenta. The threecomponents of momentum, px, py, pz which might threaten spurious cross-terms, are limited (i.e. cross-terms killed) by the introduction of matrixoperators, four-spinors; and similarly for cross-terms with β.

The linear relativistic Schrödinger equation (LRSE) [using the standard op-erators for energy and momentum] is then:iH∂ψ/∂t = Hc/i (α1∂ψ/∂x

1 + α2∂ψ/∂x2 + α3∂ψ/∂x

3) + βmc2 ≡ Hψ. Notecomparison: While the Schrödinger equation is in general chartered for de-pendence upon the momentum, for a non-relativistic free particle this isencompassed within a plane wave.

The coecients αi [and β] here cannot be numbers, since the equation wouldnot then be invariant even under a spatial rotation. Also,... the wavefunctionψ cannot be a simple scalar; rather it is a 4-component column vector. ...the probability density ρ = ψ ∗ ψ should be the the time component of aconserved four-vector if its integral over all space, at xed t, is to be aninvariant. [Is anybody disturbed about the issue of simultaneity? I wouldomit the words if its integral over all space at xed t].

To free [the LRSE] from these limitations, Dirac proposed that it be con-sidered a matrix equation. There are several constraints upon the operatormatrices and the state vector, for physical acceptability:

1. It must give the correct energy-momentum relation. That is, eachcomponent ψσ of the wave equation must satisfy the Klein-Gordonequation: −H2∂2ψσ/∂t

2 = (−H2c2∇2 +m2c4)ψσ.

2. It must allow a continuity equation and a probability interpretation forψ.

3. It must be Lorentz-invariant.

64

Matrices and vectors/spinors of no fewer than 4 dimensions suce.

ψ =

((ψ1

ψ2

)(ψ3

ψ4

)) (90)

Pursuing (1): We iterate the LRSE and inspect it, to determine the non-vanishing elements of α and β, and to apply the stated constraints:

−∂2ψ

∂t2= −~2c2

3∑i,j=1

1

2(αjαi + αiαj)

∂2ψ

∂xi∂xj+~mci

3∑i=1

(αiβ + βαi)∂ψ

∂xi+β2m2c4

(91)

We may resurrect [the LRSE] if the four matrices αi, β obey the algebra

αiαk + αkαi = 2δik, (92)

αβ + βα = 0, (93)

α2i = β2 = 1. (94)

What other properties do we require of these matrices αi, β , and can weexplicitly construct them?The αi and β must be hermitian in order that the hamiltonian [in LRSE] bea hermitian operator...Since α2

i = β2 = 1, the eigenvalues must be ±1.

Also, from the anticommutation properties [αβ + βα = 0] that the trace...of each αi and β is zero. For example αi = −βαβ and by the cyclic propertyof the trace [in general] Tr (AB) = Tr (BA)

Trαi = +Trβ2αi = +Trβαiβ = −Trαi = 0 .And from αiβ + βαi = 0 , β = −αiβαi , Trβ = −Trβα2 = −Trβ = 0.

Since the trace is just the sum of the eigenvalues, the number of positiveand negative eigenvalues must be equal, and the αi and β , must therefore be

65

even-dimensional matrices. The smallest even-valued dimension, N = 2 , istherefore ruled out, since it can accommodate only the three anticommtingPauli matrices σi plus a unit matrix. The smallest dimension in which theαi and β can be realized is N = 4, and that is the case we shall study. In aparticular explicit representation the matrices are

αi =

(0σiσi0

)and

(10

01

), where the σi are the familiar 2x2 Pauli matrices,

and the unit matrices in β stand for 2x2 unit matrices.

Pursuing (2): To construct the dierential law of current conservation, werst introduce the hermitian-conjugate wave functions ψ† = (ψ1∗, ψ2∗, ψ3∗, ψ4∗)and left-multiply the LRSE by [this row-vector] ψ† .Next we form the hermitian conjugate of [the LRSE] and right-multiply byψ . Then we subtract the latter equation from the former. The result is:

i~∂

∂t

(ψ†ψ

)=

3∑k=1

~ci

∂

∂xk(ψ†αkψ

), (95)

or∂ρ

∂t+∇·j [Equation of Conserved Probability Flow, ECPF], where we

make the identication of probability density

ρ = ψ†ψ =4∑

σ=1

ψ∗σψσ (96)

and of a probability current [PC3C] with three components

jk = cψ†αkψ. (97)

Integrating ECPF over all space and using Green's Theorem,∂

∂t

´d3xψ∗ψ =

0.

The notation... anticipates that the probability current j forms a vector if[ECPF] is to be invariant under three-dimensional space rotations. We mustactually show much more than this. The density and current in [ECP¡ F]must form a 4-vector under Lorentz transformations in order to insure the co-variance of the continuity equation and the probability interpretation. Also,the Dirac Equation [LRSE] must be shown to be Lorentz covariant before we

66

may regard it as satisfactory....

Before delving into the problem of establishing Lorentz invariance of theDirac theory, it is perhaps more urgent to to see rst that the equationmakes sense physically. [Note that these authors share my frequent issue ofdesiring to proceed in several directions at once!]

We start simply by considering a free electron and counting the the numberof solutions corresponding to an electron at rest. Equation [LRSE] thenreduces to iH∂ψ/∂t since the deBroglie wavelength is innitely large and thewavefunction is uniform over all space.

In the specic representation for α and β [given above], we can write downby inspection four solutions:

These are e−(imc2/h)t

((1

0

)(

0

0

)) , e−(imc2/h)t

((0

1

)(

0

0

)) , e(imc2/h)t

((0

0

)(

1

0

)) , and

e(imc2/h)t

((0

0

)(

0

1

)) ,

the rst two of which correspond to positive energy, and the second two tonegative energy. The extraneous negative-energy solutions which result fromthe quadratic form of... [the Klein-Gordon equation] are a major diculty,but one for which the resolution leads to an important triumph [Ch. 5 ofBDRQM]... antiparticles.

Regarding the positive-energy solutions, we wish to show that they have asensible non-relativistic reduction to the two-component Pauli spin theory.To this end we introduct an interaction with an external electromagnetic elddescribed by a 4-potential: Aµ : (φ,A) .

The coupling is most simply introduced by means of the gauge-invariantsubstitution:pµ → pµ− e

cAµ made in classical relativistic mechanics to describe the inter-

action of a point charge e with an applied eld.

The Dirac Equation modied for the eect of an EM eld is:

67

i~∂ψ

∂t=(cα ·

(p− e

cA)

+ βmc2 + eφ)ψ. (98)

This equation expresses the 'minimal' interaction of a Dirac particle, con-sidered to be a [massive] point charge, with an applied electromagnetic eld.To emphasize its classical parallel, we rewrite [DEEM] with H = H0 + H ′,with H ′ = −eα · A + eφ. The matrix cα appears here as the operatortranscription of the velocity operator in the classical interaction expression

for the interaction of a point charge: H ′classical = −ecv ·A + eφ. The oper-

ator correspondence is vop = cα. This operator correspondence , is againevident in equation [PC3C] for the probability current.

BDRQM continue to establish the upper/lower two spinor components as thelarge/small parts of the 4-component wave function.

A The Gaussian Beam; the Fundamental Ray

in Free Space

From Wikipedia: For a Gaussian beam [or ray], the complex E-eld ampli-tude is given byE(r, z) = E0(wo/w1)exp −r2/w2

1 − ikz − ikr2/(2R(z)) + iς(z).Here E0 = |E(0, 0)|; r is the radial distance from the center axis of the beam;and z is the axial distance from the waist.k = 2π/λ is the wave-number(radians/meter). w1 is the radius at which the amplitude drops to 1/e of itsaxial value. R(z) is the radius of curvature of the wavefronts. ς(z) is theGouy phase shift, an extra contribution to the phase that is seen in Gaussianbeams.

B Initial Development of the Lagrangian, Fol-

lowing GCM

The standard treatment considers a set mi of mass points belonging to thesystem, under forces other-than-of-constraint Fi. Following D'Alembert'sprinciple, we consider a set of virtual displacements δri consistent with any

68

constraints involved (constraints here taken as holonomic: such a constraintcan be expressed by a relation fri = 0 ). The δri are given as functionsof a (likely lesser) set of generalized coordinates qj which are taken asindependently variable. (There are ways of dealing with some non-holonomicsystems, e.g. with Lagrange multipliers; but we do not need these here.) Theforces of constraint do no work.

D'Alembert's principle may be written as∑

i(p•i − Fi) • δri = 0. Color will

aord a guide to the successive algebraic alterations of terms.

The 2nd term, corresponding to qj and to the the generalized force Qj, is−Qjδqj =

∑i Fi • (∂ri/∂qj)δqj.

The 1st term entails more extended eort:∑

i p•i • δri =

∑imr

••i • δri =∑

i,j [mr••i • ∂ri/∂qj] δqj.

Employing the identity# (d(uv) = udv + vdu) upon r••i ,we obtain:∑i,j d/dt [mir

•i • ∂ri/∂qj] δqj − [mir

•i ] • [d/dt(∂ri/∂qj)]δqj.

Exchanging derivatives wrt t and wrt ri on the right, and noting thatr•i = dvi/dt, this becomes

∑i,j d/dt [mivi • ∂ri/∂qj] δqi − [mivi] • [∂vi/∂qj]δqj.

With the (striking!) substitution ∂ri/∂qj = ∂r•i /∂q•j = ∂vi/q

•j , eliminating

ri entirely, in favor of vi :∑i,j

d/dt

[mivi • ∂vi/∂q•j

]δqj − [mivi] • [∂vi/∂qj]δqj

,

which in turn equals∑

j

d/dt

[∂T/∂q•j

]δqj − ∂/∂qjTδqj

.

Note that the sum on i has disappeared into T .

Recalling now that this last expression equals the 1st term of D'Alembert'sprinciple, apply the 2nd term equal to −Qjδqj, and obtain∑

j

d/dt

[∂T/∂q•j

]− ∂/∂qjT −Qj

δqj = 0.

GCM recognizes these (plural because of j) as Lagrange's equations for suchrelatively general cases as non-conservative forces.

In QM, the forces are generally conservative, i.e. derivable from a potential,so that Qj = −∂V/∂qj; and, specically, they do not depend upon the ve-locities q•j . Thus the force term can be collapsed together with T to yield∑

j

d/dt

[∂L/∂q•j

]− ∂/∂qjL

δqj = 0, where L ≡ T − V . This conveys the

more typically recognized Lagrange's equations.

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C The Sturm-Liouville Problem

(Morse and Feshbach, Methods of Theoretical Physics 1953, vol. 1, p.719)

The ordinary dierential equation arising from the separation of the generalpartial dierential equation∇2ψ + k2ψ = 0 can... be written in the form:d/dz [p(z)dψ/dz]+[q(z) + λr(z)]ψ = 0 . This is called the Liouville equation.

The parameter λ is the separation constant [or eigenvalue] (in some cases,more than one separation constant appears...). Each of the functions p, q, rare characteristic of the coordinates used in the separation......

The Sturm-Liouville problem is essentially the problem of determining thedependence of the general behavior of ψ on the boundary conditions imposedon ψ.

The solutions exhibit the features cited in the main text regarding completesets of functions permitting determination of quasi-lengths, hence normal-ization, projection; supporting Schwartz' Inequality, etc. See the referencefor further discussion.

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