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QUANTUM MECHANICS

A Co-development of Quantum Mechanics

and

Lagrangian/Hamiltonian Classical Mechanics,

with

Perspectives from Quantum Electronics

and

Allied Fields

Fausts Epistle to the Cliffordians

Walter FaustNaval Research Lab (Retired)

November 9, 2012

Contents

1 Introduction 3

2 The discovery of QM 42.1 Concepts that survive from CM (and some that notably do not) . . . 62.2 Critical features, new and counterintuitive . . . . . . . . . . . . . . . 82.3 Early Observations Altogether Unexpected, Classically . . . . . . . . 92.4 Paths in space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

1

3 Systems of QM Notation 16

4 A Comparative Exposition on CM and QM 19

5 Hamilton's Wave Equation; the Eikonal 36

6 Hamilton's Place in History 416.1 QM Development of the Eikonal; the WKB Approximation . . . . . . 41

7 Orbital Angular Momentum, Intrinsic Spin 447.1 Types of Angular Momentum . . . . . . . . . . . . . . . . . . . . . . 447.2 Quantization with respect to Rotation . . . . . . . . . . . . . . . . . 457.3 Symmetry upon Exchange of Particles; Fermi and Bose Statistics . . 467.4 Experimental Discovery and Theoretical Account for Intrinsic Spin . . 467.5 Orbital Angular Momentum; Summation over Contributions to An-

gular Momentum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 477.6 Commuting Operators for Angular Momentum . . . . . . . . . . . . . 48

7.6.1 Commutators Connecting the Coordinates . . . . . . . . . . . 497.6.2 Raising and Lowering Jz . . . . . . . . . . . . . . . . . . . . . 497.6.3 Summing Multiple Degrees of Freedom . . . . . . . . . . . . . 50

8 The Dirac relativistic theory of the individual free particle (FermiStatistics): 51

9 The Dirac Equation 53

A The Gaussian Beam; the Fundamental Ray 58

B Initial development of the Lagrangian, following [GCM] 58

C Sturm Louisville Problem 60

D Mathematical Foundations 60D.1 Hilbert Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60D.2 Hermitian Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . 61D.3 Dirac Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

2

References

[GCM] Goldstein, Classical Mechanics

[LLQM] Landau and Lifshitz, Quantum M echanics

[DVQM] Davydov,Quantum Mechanics

[DWIQM] Dicke and Wittke Introductory Quantum Mechanics

[CTQM] Cohen Tannoudji Quantum Mechanics

[SQM] Schi, Quantum Mechanics

[GTQM] Heine, Group Theory in Quantum Mechanics

[HRSP] Hehre, Radom, Schleyer, and Pople, Ab Initio Molecular Orbital Theory

[CS] Condon and Shortley, Theory of Atomic Spectra

[PDF] Attached pdf les

1 Introduction

Classical mechanics (CM) is appreciated in a relatively intuitive fashion. Darwin'sagencies, over millions of years, have organized our brains to deal with this mattereciently as a lady robin knows how to build a nest without having previously seenher mother build one.

But microscopes and accelerators are very recent inventions, so QM (quantum me-chanics) and SR (special relativity) don't come so naturally; GR remains a mysteryto most of us (general relativity; black holes represent an even more recent discov-ery). [GCM], incidentally, remarks that QM requires a much more violent recastingthan does SR. These recastings are found necessary, respectively, as material objectsget tiny and/or move very fast (and nally, in intense gravitational elds).

We benet from a succession of surges in human understanding, apparent successes.Regarding the associated brilliance: The innate gifts, the labors, of those who havemade the great strides certainly are not to be minimized. But one's chancy arrival

3

at a particularly productive perspective deserves emphasis as well; and this suggestshumility for most of us.

2 The discovery of QM

Historically, the entry into QM came through insights re the corpuscular nature oflight: Planck's treatment of black-body distribution (1900); Einstein's treatment ofthe photoelectric eect (1905); and Compton's treatment of x-ray scattering fromatoms (1923).

From another perspective, we had not fully arrived at QM until a wavelength wasassociated with the propagation of material particles; they emulate light. De Broglie(1924) rst associated wavelike properties, hence a wavelength = h/p, with motionof material objects (p the momentum).

The bridge between QM and CM is a narrow one. One indication is that many bookson the one topic scarcely refer to the other. And it develops that the applicable skills,the methodologies, are quite dierent.

From a modern perspective, Erwin Schrdinger may seem to have faced a staggeringchallenge akin to reverse engineering: to construct a microscopic model on the basisof macroscopic and limited microscopic information. The macroscopic model of CMhad already been brought to maturity, by Lagrange (1788) and Hamilton (1833), butin retrospect it was quite far removed from the ultimate microscopic model of QM.It is now appreciated that the assertions that ow, for a given dynamical system,from a valid QM model become indistinguishable from those of CM in the limit(from the CM view) that the action S becomes large relative to Planck's constanth ' 6.61027ergsec; or (from the QM view) in the limit of large quantum numbers.(Terms will be further dened as we progress). But this point, taken alone, aordslittle insight into Schrdinger's problem.

On the more supportive side, indications had arisen before Schrdinger, that themotion of matter manifests the character of wave propagation, hence that that thereis an associated wavelength.

Reports in this line, from Wikipedia:

1. The Old Quantum Theory of Bohr and Sommereld (1913/25) had laid out

4

integral numbers of wavelengths for an electron tracing a Kepplerian orbitabout a nucleus, accounting for discrete electronic energy levels.

2. de Broglie (1923/4) had associated the wavelength of a free particle with itsmomentum , as = h/p :

{Following Planck's E = h and Einstein's E = mc2, de Broglie equated thetwo, but withmc2 replaced, for material particles, bymv2, to obtain h = mv2.Replacing by v/ , (again adopting v instead of c for the velocity of a materialparticle), he obtained hv/ = mv2 or h/ = mv = p , rearranged to the now-familiar = h/p.}

3. Schrdinger (1926) published an equation describing how the matter waveshould evolve (in time), the matter wave equivalent of Maxwell's equations,and used it to derive the energy spectrum of hydrogen.

Concepts due to Schrdinger were central, essential, and far-reaching:

(a) The wave function of a system: a complex-number function (conveyingamplitude and phase) of the specic state in which the system is foundat a given instant. Thus

1 = i and complex algebra enter. Phase

naturally enters into the wave function for wave propagation or for theevolution of an entangled system (localized or an extended). The sense issimilar to that in electronics.

However, in electronics we write a voltage function of time as V eit, thenretain only the real part |V |cos(t). QM diers in that any complexquantity gets multiplied by its complex conjugate, or with a Hermitian,ie a real, operator M sandwiched between, to evaluate an observablequantity. This again yields a real number, but thus in a dierent fashionfrom electronics.

When is an eigenfunction of each of a set of commuting operators {M},the state often is written to display explicitly the corresponding value ofeach operator.

Since the system must show up somewhere in the space or hyperspaceinvolved (for the present we assume perfect detectors), the probabilitiessum or integrate to unity:

(q) (q) dq = 1, where the variables q span

the space. Then (q) = (q) (q) denes the probability density and (q) dq = 1.

5

(b) A set of operators characteristic of the system, such that when each isexercised with , it conveys the value of an observable characteristic ofthe system. More specically, when the operator is applied in ,the result represents an average nding for the corresponding variable,given that state . This is written as = (for now may betaken as the conjugate).

Apart from the position operators x, y, and z these operators typicallyare partial derivatives (henceforth we use ~ for h/2 ):

The energy i~

t;

The x -component of linear momentum i~ x

.

The z-component of angular momentum i~ z

.

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

1. The wavefunction is burdened with all the information that can be knownabout the system, even in principle.

Conversely: A function which accommodates every measurable property of asystem is automatically complete in this sense. This is a critical point towardDirac's 4-component model for the SR electron.

2. Superposition, interference, and entanglement. These are developed at length,below. Schrdinger evidently understood these quite well, cf his tale of Schrdinger'sCat; he even employed the German word for entanglement, vershrankung.

3. Canonically conjugate observables, and their constraint by the uncertaintyprinciple.

2.1 Concepts that survive from CM (and some that notablydo not)

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

6

ences; and the state of this system.

Observable variables associated with a state, such as position r, momentum p, energyE - which, beyond the very constituents of the s