Answers to Selected Exercises

Answers or hints are provided here for most of the exercises given at the end of chapters. Answers proposed are not necessarily unique.

2.1 a) ((hl1>2) = 1, (1)211>3) = -i, (1)311>1) = o. b) C1 = 1/)5, C2 = 1/)2, C3 = 1.

2.2 b) a = 1/y3, (3 = 1/)2. c) 11>3) = (1/V6) {2ilu1) -lu2) + IU3)}.

2.3 1(1)1?)!)1 2 = 41 < 231 = (1)I1>)(?)!I?)!).

2.4 To prove that equality implies proportionality, write 11» = .xl?)!) + la), where (al?)!) = 0, and show that la) = O.

2.5 ° is not linear and does not have an inverse; P is linear but does not have an inverse.

2.6 a) Evaluate (ot {a111>1) + a211>2)}, I?)!)) for aI, a2, 11>1)' 11>2) and I?)!) arbi-trary. b) Evaluate (11)), (ot) t I?)!)). c) Evaluate ((010 2) t 11», I?)!)).

2.7 (1)IOI?)!) = i - 7 = (?)!IOI1»*.

2.8 Evaluate (?)!I {11»(¢I} Ix) for I?)!) and Ix) arbitrary.

2.9 Evaluate [U]t [U].

2.10 a) Evaluate (ukILluz). b) £kZ = Lj LkjLjz. 2.11

11>1) = (1/2) { -V2ilul) - ilu2) + IU3) } ;

1¢2) = J273 {(i/V2)lu1) + IU3)};

526 Answers to Selected Exercises

2.12 a)

2.13 a)

2.14

2.15

14>1) = (1/v'2) {ilul) + IU3)};

14>2) = IU2) ;

14>3) = (1/v'2) {-ilul) + IU3)}.

14>1) = (1/2) { -v'2ilul) + IU2) -IU3)};

14>2) = (1/2) {v'2IU1) - ilu2) + ilu3)};

14>3) = (1/v'2) {IU2) + IU3)}.

14>1) = (1/v'2) { 1:llul) -IU2) } ;

14>2) = (1/v'2) {1:llul) + IU2)} .

14>1) = (1/v'2) {lUI) -IU3)};

14>2) = (1/2) {lUI) + v'2IU2) + IU3) } ;

14>3) = (1/2) {lUI) - v'2IU2) + IU3)} .

2.16 Direct verification.

2.17 Direct verification.

2.18 a) Check the formula for n = 1, and differentiate the formula for arbitrary n-l. b) Expand F(1) in a Taylor series around t = O.

2.19 The lemma is true if B is unitary. It is not true in general, as can be seen by letting

Answers to Selected Exercises 527

2.20 The matrices make up a complete set of commuting operators. Here are common eigenvectors:

( 1 ) (1) (0) ( 0 ) 1 -i 1 i 1 0 1 0

V2 ~' V2 ~' V2 ~' V2 ~1 .

2.21 a) Common eigenvectors are:

b) {[H], [K]}.

3.1 a) Possible results: -1, 1,2; probabilities: 1/4, 1/4, 1/2. b) (A) = 1; LlA = J372. 3.2 b) P(A ----> 0) = 1/6; P(A ----> 6) = 5/6; (A) = 5; LlA = J5. 3.3

Var(H) = ICl 12Ei + IC212E~ _ {IClI2E1 + IC212E2}2 IC1 12 + IC212 ICl 12 + IC212

IC1 121C212 (E _ E )2 {ICl I2 + IC212}2 1 2·

3.4 I1/>( t)) = exp( -iEt/ h) {cos( at/ h) lUI) + i sin( at/h) IU2)}.

3.5 a) 11/>(t = 0)) = (l/V2) {I U2) - IU3)}. b) P(K ----> -b;t) = 1.

3.6 a) 11/>(t)) = (1/V2){exp( -iEt/h)lul) - exp(iEt/h)lu2)}. b) (1/>(t)IKI1/>(t)) = b/2.

3.7 b) 11/>(t)) = -(1/V2) exp( -8it/h)IE1 ) + (1/V2) exp(4it/h)IE3). c) (H) = 2; LlH = 6.

528 Answers to Selected Exercises

3.8 a) T/I = 1, IT/I) = lEI); T/2 = 1, 1T/2) = IE2); T/3 = 0, 1T/3) = IE3). c)

1 (2 -v'2 v'2) [KJ = 4 -v'2 3 1 . v'2 1 3

3.9 b) l"p(t)) = (1/v'2) exp( -iE't/n) {exp( -ibt/n)IE2 ) + exp(ibt/n)IE3)}. c) (H) = E'; AH = Ibl.

3.10 a) 2, 1, -l. b) P(A -t 2; to) = 1/2. c) (A) = 1 d) Several possibilities.

4.2 b) Set o:A = iOO[lyJ and {JB = ioO[lxJ in (a).

4.3 Note that (1x'!y,!z) is irreducible if and only if (1+, L,!z) is.

4.9 a) Let e and 4> be the polar and azimuthal angles of u. Then

2 e.:1, 1 - 2 e .:1, 11; u) = cos 2 e- ' 'I' I 1) + v'2 sin 010) + sin 2e''l'1-1),

10;u) = - ~sinee-i¢11) +coseIO) + ~sineei¢I-1),

2 e.¢ 1 - 2 e .¢ 1-1;u)=-sin 2 e- ' 11)+ v'2sinOIO)-cos 2e' 1-1).

b) 1/4, 1/2, 1/4.

5.1 Integrate by parts to obtain the first identity. To obtain the second one, take f(x) = F(x)(x - xO)2.

5.2 L1c; can represent a delta function, but not its derivatives.

5.3 Evaluate J dp lip) (pil where I is represented as in (5.1).

5.4 Evaluate the (xx') matrix element of (5.18) and use properties of the delta function.

Answers to Selected Exercises 529

5.5 First show, by induction, that (xlpn l7{i(t)) = (-intxf 7{i(x, t). To obtain the second equality, use the Fourier transform.

5.7 (7{iIXI7{i) = 0 = (7{iIPI7{i)· 5.8

{ II }-1/2 { }1/2 A = a + -k sin(2ka) + - cos2 (ka) = -/'\,-

2 /'\, 1 + /'\,a

5.9 -0.9796; -0.8176.

5.11 For x > a write 7{i(x) = A+eikx + B+e- ikx , where k2 = 2mEjn2.

5.12 c) (k-k')2j(k+k')2.

5.14 Substitute functions in the differential equation, with appropriate va-lues of E.

5.15

5.16

if m = n + 1, if m ¥ n ± 1, if m = n - 1.

c) ~X = {(2n + 1)nj2wm}1/2, ~p = {(2n + 1)mnwj2}1/2.

5.19 a) Use the definition of the propagator and properties of the evolution ope-rator. b) Replace U by its value and carry out the integral.

5.20 Use properties of the evolution operator.

6.1 The solid angle corresponding to Az > 0 is 27l'. The region where ~ is 1 is proportional to 7l' - e', whereas the region where ~ is -1 is proportional to e'. 6.2 Expand 7{i(xo, to) in terms of harmonic oscillator eigenfunctions and pinpoint the most important coefficient.

7.1 b) Express (¢IHI7{i) - (7{iIHI¢)* as the integral of a divergence, and transform it into a surface integral.

7.7 Separate terms that depend on T from those that depend on e and ¢.

530 Answers to Selected Exercises

7.10 6.R = V3ao/2Z.

7.11 b) (R2) = 3a6/Z2, (p. P) = (nz/ao)2, 6.X6.Px = n/V3.

7.12 E = nw {nx + ny + nz + 3/2} where nx, ny and n z are nonnegative integers.

7.14 b) Estimate the mean value of Ck and the mean value of q(R· B)Xk in a state with typical atomic size. The ratio of the latter to the former is of order 10-5 .

7.16 Evaluate ft 1~12 by means of the Schrodinger equation for a particle in an electromagnetic field.

8.2 Write ~(~) = f (0 exp{ ±Fc 0 and show that the equation for f( 0 has a solution with leading term ~n only if n = O.

8.4 b) Show that d~/d~ is an odd function and use (a).

9.2 AH / AD = 1.000272; AH / AT = 1.000362.

9.3 Approximately 4.

9.4 a) The equation separates in cartesian coordinates and the solutions are products of sines. b) See what the possible values are for nx , ny, n z .

9.6 a) A = 144, n = 3.

9.7 a)

[! 1 {2ZcfJ }3/2 qe - 3n2 ao(3~

9.8 The first integral is bounded and the second one vanishes if r is large enough.

10.1 Holxk) + Win-I) = 2::;=0 cpIXk-p)'

10.2 Project (10.11) on (Efl, where j -=I- n.

Answers to Selected Exercises 531

10.3 Corrections of eigenvalues vanish. Corrections of vectors lEI) and IE2 )

are different from zero.

10.4 a) In the degenerate sector one has

b) Solve the secular equation approximately, keeping second-order terms in a and b. Eigenvalues then coincide with those found by perturbation theory.

10.5 c) 51.

10.7 b)

(300IWI31O) = (310IWI300) = a;

(31OIWI320) = (320IWI31O) = b; (311IWI321) = (321IWI311) = c

= (31-1IWI32-1) = (32-1IWI31-1).

c) Order rows and columns suitably; the matrix breaks down into blocks with dimensions not larger than 3 x 3. d) Three levels are unchanged; two levels are shifted by +c, two more by -c; the last two are shifted by ±va2 + b2 .

10.8 Use the binomial expansion up to second-order terms.

10.9 Pick f = z and make use of spherical coordinates.

10.11 The value computed up to second order coincides with the exact value.

10.12 Write X 3 and X 4 in terms of creation and annihilation operators.

11.3 Write down matrix elements of equation (>.J - H)G(>") = I. 11.5

F(8) = 3Z 3 {sin(al~kl) -al~klcos(al~kl)}. (al~kl)

11.6 a) F(8) = Z{l + a21~kI2} -2 .

532 Answers to Selected Exercises

11.7 a) f(B, ¢) = -(2m/n2 )Voa3 {1 + a21~kI2} -1 .

11.9 a) 'iil(r) = rjv(kr) where v(v + 1) = l(l + 1) + A. b) cSl = (1[/2)(l- v).

11.11 Write (11.77) in terms of hi and hI' 11.12 b) Write ~k = qaA + qbB + qcC + D and show that D = O.

12.1 a) One can write at any time t,

in:t l1j;(t))-HI1j;(t)) = LCilui), t

where {lUi)} is an orthonormal basis containing 11j;). Show that all coefficients but one vanish.

12.4 a) (1/2)1. b) Same result.

12.5 b) All mean values vanish except Tr(LJ10"y) = 1.

12.6 Use the definition of derivative.

12.7 Show that each subsystem obeys the equation given in Exercise (12.1a) and use the result of that exercise.

12.8 Expand IX(l, 2)) as in (12.26), picking the lUi) and IVk) as eigenvectors of H(l) and H(2) respectively.

12.11 a) Evaluate Tr{ALJoAt} in a basis where LJo is diagonal. b) Use properties of the trace and of the evolution operator.

12.13 Note that LJo = (I + u· no)/2, P{ = (1 + u . nl)/2, etc.

13.1 Invariance under translations or space inversion is straightforward. To show that the Laplacian is invariant under rotations, find the transformation

Answers to Selected Exercises 533

of (} /(}Xi. Invariance of the delta function is proved by integrating the product of the right-hand side and an arbitrary function f(r).

13.3 b) The representation is reducible.

13.4 Show that if a subspace is invariant under the action of all matrices M(g), so is its orthogonal complement.

13.5 C2v , D<Xlh, C3v '

13.6 b) D4 .

c) D4h .

13.8 c) Reckon constraints following from the definition of an orthogonal matrix with unit determinant.

13.10 c) The condition on the determinant reduces the number of parameters by one, in contrast with orthogonal matrices.

13.12 b) Write M as

1 { T 2 } 1 { T} 1 - M + M - -Tr(M)I +- M - M +-Tr(M)I. 232 3

13.13 b) [T1' T2l = 2iT2' [T2' T3l = iT1' [T3, T1l = 2iT3· 13.18 a) The unit element is (I,O) and the inverse of (O,a) is (0-1, -O-la ).

14.1 Write the constraints on matrix elements explicitly; solve the resulting equations, keeping in mind the form of O(a,{J,')').

14.4 Expand binomials and rename summation indices.

14.6 a) Carry out the product of matrices and identify the appropriate element on both sides of the equation. b) Evaluate Dho(a, (J, ')').

14.7 Use the Levi-Civita symbol.

14.8 Write 8 1 .82 in terms of 8 2 and I.

14.12 Write J 1 . J 2 in terms of J1+, J1-, etc.

534 Answers to Selected Exercises

14.15 Note that 4j is even and that j - Tn is an integer.

14.19 The operator of rank 2 is like (14.90) for PI = P 2 .

14.21 b) Equation L i ,¢ = 0 implies that '¢ only depends on r; use this result to solve the other equation.

15.2 Note that if A = -iI: j aj Eoj , then A3 = a2 A. Moreover, aiaj j(a2) =

viv j j(v2 ).

15.3 a) Note that (I: j aj (TOj)2 = -a2 I.

15,4 Use the identity (15.80).

15.9 Make use of the fact that 82Xl/8t2 rv (-iEjfi)8xl/8t, after justifica-tion.

16.1 b)

1)x = lim [2nifiEV (x + xn )] -1/2 JV(x) n-->oo 2

x 11 {[ 2nifiEV ( Xk +2Xk - 1 ) ] -1/2 dXk } .

16.3 Assume (16.29) is true for p - 1, and carry out the integral on dxp.

16.5 b) Convince yourself first that ().~1) - ).~2))j().~1) + ).~2)) --+ 0 as n --+ 00.

16.6 Show that the last two lines in (16.51) coincide if (16.49) is substituted in them.

16.9 EjVo = -0.8093; -0.6032; -0.4423; -0.3020; -0.1747; -0.0566.

16.10 There are an infinite number of bound states only if the value of n corresponding to a null value of E is infinite.

16.12 a) Use the fact that ,¢(x) = 0 for x < o. 16.13

E = {~ (n _ ~) nfi_1 Vo }2/3 2 4 y'2ma

Answers to Selected Exercises 535

17.1 The correction term is 1]2{(¢1IHI¢1) - E}.

17.2 The variational method yields Eo :::: - Vo/32.

17.3 a) Eo :::: Iiw / J2. Note that the trial function has a discontinuous derivative at x = O. Its second derivative has therefore a term proportional to 8(x). b) Eo:::: 1iw/2.

17.4

E . '" ~ ~ (3). 2m) 1/3 mm 22m 8 li2

17.9 Examine the equation obtained by multiplying (17.47) and uk(rd and by integrating on dr1'

17.11 Write the atom's energy and the ion's energy by means of (17.41).

17.13 Use (14.82) to compute the product of two spherical harmonics or Legendre polynomials.

18.7 Here the second term in (18.25) is absent.

18.13 One can use either IT; L, S; ML , Ms) or IT; L, S; J, M) as unperturbed eigenvectors, but the second choice may be simpler.

19.2 a)

b) Only the probability that the atom is in state (2p, m = 0) does not vanish.

19.6 In the matrix element, introduce the factor n- 1n, where n is the operator of a rotation by 7r about ij.

19.7 103 W m-2 .

19.10 Use the result of Exercise (14.18).

19.11 (217 W~m e2a6) / (311 lic3 ).

20.6 Note that ie transforms ra into -rb and rb into -ra; a reflection in a plane that contains the axis leaves ra, rb, Ba and Bb invariant.

References

Numbers in square brackets after each entry refer to sections where books and papers are mentioned. An asterisk indicates more than one reference to a book or paper in the same section. P and UC mean "Preface" and "Units and Constants" while lIe, for instance, means "Exercises in Chap. 11." Several monographs not specifically referred to are given as general references.

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Quantum Theory," Phys. Rev. 115, 485 (1959). [7.8] 3. H. C. Allen, Jr., P. C. Cross: Molecular Vib-rotors (Wiley, New York 1963).

[20.6] 4. H. L. Anderson, ed.: Physics Vade Mecum (American Institute of Physics,

New York 1981). 5. R. A. Arndt, L. D. Roper: "Scattering in a Spherical Potential: Motion of

Complex-plane Poles and Zeros," Compo in Phys. 3 (Marchi April), 65 (1989). [11.8]

6. A. Aspect, J. Dalibard, G. Roger: "Experimental Test of Bell's Inequalities Using Time-varying Analyzers," Phys. Rev. Lett. 49, 1804 (1982). [21.2,21.3]

7. A. Aspect, P. Grangier, G. Roger: "Experimental Tests of Realistic Local Theories via Bell's Theorem," Phys. Rev. Lett. 47, 460 (1981). [21.2]

8. L. E. Ballentine: "The Statistical Interpretation of Quantum Mechanics," Rev. Mod. Phys. 42, 358 (1970). [6.2]

9. L. E. Ballentine: "Probability Theory in Quantum Mechanics," Am. J. Phys. 54, 883 (1986). [3.2]

10. L. E. Ballentine: "Resource letter IQM-2: Foundations of Quantum Mechanics Since the Bell Inequalities," Am. J. Phys. 55, 785 (1987).

11. L. E. Ballentine, ed.: Foundations of Quantum Mechanics Since the Bell In-equalities, Selected Reprints (American Association of Physics Teachers, Col-lege Park 1988).

12. L. E. Ballentine: "What Do We Learn About Quantum Mechanics from the Theory of Measurement?" Int. J. Theor. Phys. 27, 211 (1988). [21.5]

13. L. E. Ballentine: Quantum Mechanics (Prentice-Hall, Englewood Cliffs 1990). [5.9.2, 6.3, 13.6, 21.2]

14. L. E. Ballentine, J. P. Jarrett: "Bell's Theorem: Does Quantum Mechanics Contradict Relativity?" Am. J. Phys. 55,696 (1987). [21.3]

15. C. N. Banwell: Fundamentals of Molecular Spectroscopy, 3rd edn. (McGraw-Hill, London 1983). [10.7, 19.6]

538 References

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17. J. L. Basdevant, J. Dalibard: The Quantum Mechanics Solver. How to Apply Quantum Theory to Modern Physics (Springer-Verlag, Berlin 2000).

18. J. L. Basevant, J. Dalibard: Quantum Mechanics, including a CD-ROM by M. Joffre (Springer-Verlag, Berlin 2002).

19. A. O. Barut, R. Raczka: Theory of Group Representations and Applications, 2nd edn. (Polish Scientific Publishers, Warsaw 1980). [13]

20. J. L. Basdevant: Mecanique quantique (Marketing, Paris 1986). 21. W. E. Baylis, G. W. F. Drake: "Units and Constants." In: [71], pp. 1-5. [UC*] 22. J. S. Bell: "On the Einstein Podolsky Rosen Paradox," Physics 1, 195 (1964).

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Rev. Mod. Phys. 38, 447 (1966). [6.2] 24. M. V. Berry: "Quantal Phase Factors Accompanying Adiabatic Changes,"

Proc. R. Soc. London, Ser. A 392, 45 (1984). [7.8] 25. H. A. Bethe, R. W. Jackiw: Intermediate Quantum Mechanics, 2nd edn.

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Atoms (Springer-Verlag, Berlin 1957). [19.2] 27. L. C. Biedenharn, J. D. Louck: Angular Momentum in Quantum Physics.

Theory and Applications (Encyclopedia of Mathematics and its Applications, Vol. 8; Addison-Wesley, Reading 1981). [14]

28. L. C. Biedenharn, J. D. Louck The Racah- Wigner Algebra in Quantum Theory (Encyclopedia of Mathematics and its Applications, Vol. 9; Addison-Wesley, Reading 1981). [14]

29. J. D. Bjorken, S. D. Drell: Relativistic Quantum Mechanics (McGraw-Hill, New York 1964). [15, 15.5]

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31. J. M. Blatt: "Practical Points Concerning the Solution of the Schrodinger Equation," J. Compo Phys. 1, 382 (1967). [8.2]

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33. A. Bohm: Quantum Mechanics. Foundations and Applications, 3rd edn. (Springer-Verlag, New York 1993).

34. D. Bohm: Quantum Theory (Prentice-Hall, Englewood Cliffs 1951). [21.1] 35. D. Bohm: "A Suggested Interpretation of the Quantum Theory in Terms of

'Hidden' Variables" I and II, Phys. Rev. 85, 166, 180 (1952). [6.2J 36. N. Bohr: "Can Quantum-mechanical Description of Physical Reality be Con-

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Phys. Today 44 (October), 36 (1991). [2l.6]

Index

The symbol "n" following a page number refers to a footnote, while the symbol "e" refers to an exercise.

A Abelian, 280 absorption of radiation, 456 action, 363, 365 active transformation, 20, 37 Aharonov-Bohm effect, 151 a decay, 88, 384 a particle, 5, 225 aluminum, 200-201, 432 ampere, XV analytic continuation, 231, 367 analytic function, 292 angstrom, XV angular frequency, 7, 89 angular momentum, 5, 6, 33, 53, 309,

319, 327 - orbital, 134-138, 148, 415 - total, 434, 472 annihilation operator, 90 antibonding orbital, 479 antilinearity, 12, 276n antisymmetric molecular function, 493 antisymmetric wave function, 396-398 apparatus, 31, 33, 111, 116, 122,502,

509 argon, 200-201, 432 arsenic, 200-201 associated Legendre function, 137, 152 associativity, 277 atom, 1-2, 31, 41, 53, 110, 183, 183n,

225, 404, 413, 437 - alkali, 5, 462 - exotic, 213, 222e atomic form factor, 237, 252, 253e

atomic radius, 2, 5, 183 atomism, 1 average energy of configuration, 422,

433 Avogadro's number, 2

B Balmer formula, 5 basis of vectors, 12 - orthogonal, 13 - orthonormal, 13, 33, 44 Bell inequality, 504 Bell theorem, 504 beryllium, 194, 200-201, 432, 520 Bessel function, 156, 378 - spherical, 143, 157, 240, 242 blackbody, 6 Bohr magneton, XV, 65 Bohr model, 5 Bohr radius, XV, 187 Bohr-Sommerfeld rule, 381 Boltzmann constant, XV, 6 bonding orbital, 479, 485 Born approximation, 235-238 Born-Oppenheimer approximation, 472 boron, 194, 200-201, 432 bound state, 130 boundary conditions, 78, 170, 173 box normalization, 81, 170, 177, 191,

451 bra, 13, 101 Breit-Wigner formula, 248 Brillouin's theorem, 405, 442 bromine, 200-201

550 Index

c C 2h , 299 Coo, 287, 481 Cooh, 287 Coov , 287, 481 Cn, 287 Cnh,287 Cnv , 287 calcium, 200~201 canonical distribution, 266 canonical momentum, 146 carbon, 194, 200~201, 430, 432 cavity, 6 center of mass, 185 center-of-mass frame, 227 centimeter, XVI central-field model, 189, 193 cesium, 4 CH4 ,493 character, 285 characteristic function, 386 charge, 33 charge conjugation, 347 charge density, XVI, 145 chemical bond, 485 chlorine, 200~201, 432 chromium, 200~201 circumflex, 55n class, 281 classical determinism, 386 classical field, 31 classical mechanics, 1, 6, 31, 110, 124,

128, 151 classical object, 384 classical particle, 88 classical physics, 1, 9 classical property, 385 classical state, 385 Clebsch~Gordan coefficients, 321,

323~326, 333e closed shell, 196 closure, 277 closure relation, 19 CO2 ,300e cobalt, 200~201 coherent state, 97 cold emission of electrons, 384 collapse, 117, 513

collective coordinate, 384, 519 commutativity, 280 commutator, 25, 44 ~ of X and P, 75, 128 complementarity, 122 complete set of commuting operators,

26, 140 complete theory, 499, 507 composite system, 260 compound,2 Compton effect, 8 configuration, 193, 414 ~ conjugate, 421, 428 configuration interaction, 441, 485 configuration space, 365 conjugate configuration, 421, 428 conjugate momentum, 384 connected group, 308n connection of WKB solutions, 379 consistent family of histories 269 274e

386, 521 ' , , continuity equation, 132, 347 continuous spectrum, 79, 87, 130, 170,

177, 228, 449 contravariant four-vector, 337, 341 convergence, 100, 235 ~ of path integral, 367 convergence factor, 236, 368 coordinate representation, 76, 77, 138 Copenhagen interpretation, 123, 514 copper, 200~201 correlation, 503 ~ electronic, 189, 441 coset, 280 covalent bond, 485 covariant four-vector, 338, 341 creation operator, 90 crisis, 9 cross section ~ differential, 226, 229, 239, 243, 251 ~ total, 226, 241, 243, 248 crystal, 31, 298 cube, 287 cubic, 299 current density, XVI, 145

D

D4 h, 299 D6h, 299 Doo ,287 Dooh, 287, 481 D n ,287 Dn d,287 Dn h,287 Dalton's law, 2 de Broglie wavelength, 8 decoherence, 5] 9 degeneracy, 22, 86, 140, 188, 188n, 209,

449 - removed, 210, 436, 437 delta (Kronecker), 17 delta function, 72, 97-100, 104e density of states, 191, 449 density operator, 256, 257, 266 determinant of an operator, 20 deterministic theory, 505 deuterium, 187, 202e diagonalization of a matrix, 23, 25 diatomic molecule, 218, 481, 497e differential equation, 39, 81, 192 diffraction, 8 dimension, 33, 38, 72, 73n, 227 - of dual space, 13 - of Lie group, 292 - of representation, 281 - of state space, 32 - of vector space, 12 dipole moment - electric, 214, 215, 217, 489 Dirac delta function, 72, 97-100, 104e Dirac equation, 343, 346 direct product, 280 direct sum, 284, 319 - of operators, 26, 59 - of vector spaces, 26, 59 discrete spectrum, 79, 81, 86, 89, 130,

139, 142, 143, 167, 174, 206, 392, 449,453

discrete variable, 163 discreteness of energy, 4-8 dissipation, 519 dissociation energy, 48 DNA,493 dodecahedron, 287 domain of operator, 101

Doppler effect, 465 double refraction, 3 dual space, 13

Index 551

duality (wave-particle), 8 dynamical variable, 33, 41, 55, 148,

149, 457, 500 - associated with projector, 37 - well-defined, 42, 44, 45

E effective potential, 139, 384 Ehrenfest relations, 125, 149, 384 eigenfunction, 79 eigenvalue, 16 - degenerate, 22, 35, 282 - of Hamiltonian, 38, 42, 79, 230 - of Hermitian operator, 16, 21-23, 33 - of momentum operator, 74 - of position operator, 73 - of projector, 17 eigenvalue equation, 16, 42, 79, 129,

185 eigenvector, 16 - of angular momentum, 137 - of Hamiltonian, 38, 42, 79, 129 - of Hermitian operator, 16, 21-23 - of momentum operator, 73, 101, 128

of position operator, 73, 101, 127 of unitary operator, 16

electric charge, 33 electric constant, XV electric dipole moment, 214, 215, 217,

489 electric dipole transition, 459, 488 electric field, XVI, 5, 145, 213, 384 electric quadrupole transition, 464 electromagnetic field, 3, 38, 41, 97, 110,

145-147, 346, 349, 450, 457 - quantization, 185, 351, 358 electron, XV, 4, 5, 7, 45, 64, 110, 183,

225,351 electron volt, XV emission of radiation, 5 - spontaneous, 457, 462, 468e - stimulated, 456 energy, 33, 38, 75, 149, 183 - average, 422, 433 - correlation, 44] - discrete, 4--8

552 Index

- dissociation, 48 - electronic, 5, 189n, 194 - excitation, 199, 432 - Fermi, 191n, 407 - ground state, 392, 483, 493 - ionization, 189n, 198, 432 - kinetic, 183 - magnetic, 54, 184 - negative, 346 - of atom, 199n - of electromagnetic wave, 455 - of multiplet, 427 - of rotation, 221, 475, 487, 488 - of vibration, 219-220, 475, 487, 488,

494 - potential, 184 - relativistic, 185, 353 ensemble, 111, 115 entropy, 266 environment, 517 equation - differential, 39, 81, 192 - finite difference, 164 - integra-differential, 195 equipartition of energy, 6, 7 equivalence relation, 281 equivalent descriptions, 38, 275, 276n equivalent orbitals, 189 equivalent representations, 281 ether, 3 Euler angles, 68, 292, 306 evolution operator, 40-41, 43, 82, 131,

230, 267 exchange charge density, 403, 405 excitation energy, 199, 432 excited state, 47 exclusion principle, 190, 289, 346, 396 exotic atom, 213, 222e exponential of an operator, 24 exponential parametrization, 293 exponentiation, 307

F fact, 522 faithful representation, 281, 308 family of histories - complete, 268, 273e - consistent, 269, 274e, 386, 521 - enlarged, 268

Fermi energy, 191n, 407 Fermi golden rule, 450 Fermi-Dirac statistics, 192 fine structure, 5, 357 fine structure constant, XV, 357 finite group, 284-290 finite-difference equation, 164 finite-element method, 173n fluctuation, 519 fluorine, 195, 200-201, 432 flux of particles, 133 forbidden transition, 459 force, 1 - magnetic, 54 Fourier transform, 76, 99, 236 free particle, 83, 94, 193, 368 friction, 519 fundamental state, 47 fundamental transition, 490

G gallium, 200-201 gas, 2, 8, 249 gas diffusion, 2 gauge invariance, 145, 148 gauge transformation, 145, 148-149,

348 Gaussian units, XVI, 184n Gay-Lussac's law, 2 generating function - of Hermite polynomials, 102 - of Laguerre polynomials, 155 - of Legendre polynomials, 151 generator, 293, 294, 307 - of Lorentz group, 339 - of rotation, 57, 59-62, 309 - of tensor product, 296 - of translation, 297 germanium, 200-201 GL(2; R), 302e GL(n; C), 295 GL(n; R), 295 Gleason's theorem, 258 golden rule, 450 gram, XVI Green's function, 233, 252e Green's operator, 230-233 ground state, 47 group, 277

- finite, 284-290 - Lie, 292-296 - Lorentz, 337 - of permutations, 284, 289-290 - of rotations, 278, 292, 306, 312 - of translations, 278, 292, 296-299 - point, 284, 286-287, 474, 493 - space, 299 group representation, 281n gyromagnetic ratio, XV, 351

H H2, 483, 490 H20, 300e, 493 Hamilton's equations, 125n, 386 Hamiltonian, 38 - classical, 123, 363, 386 - of atom, 185 - of atom in magnetic field, 64 - of classical object, 385 - of free particle, 83 - of hydrogen molecular ion, 46 - of identical particles, 289 - of particle in electromagnetic field,

146 - of particle in one dimension, 75, 77 - of particle in three dimensions, 128 - of total system, 260, 320 - perturbed, 206, 446 - real, 291 - relativistic, 351, 353 - time-dependent, 44, 445 - time-independent, 41, 43, 44, 391 - unperturbed, 206, 446 hard sphere, 243 harmonic oscillator, 88-93, 95, 106e,

123, 125e, 219, 223e, 302e, 329, 373, 388e, 494

- classical, 388e - in three dimensions, 160e hartree, XV Hartree equations, 195, 404n Hartree-Fock equations, 403, 475 - restricted, 406 Heaviside step function, 100 Heisenberg picture, 41 Heisenberg uncertainty principle,

119-122

Index 553

helium, 6, 194, 200-201, 384, 393, 398, 4lle, 432, 441

Hellmann-Feynman theorem, 474 Helmholtz free energy, 266 Hermite polynomial, 92, 101-104, 106e hermiticity, 15, 77, 101 - of momentum operator, 105e hexagonal, 299 hidden variables, 112-ll4, ll6, ll7,

502 Hilbert space, 32, 100, 260 history, 267, 521 - probability of, 268 HLi+, 48 hole, 422, 428 homomorphism, 279, 308 Hund's rule, 400 hydrogen, 200-201 - atom, 5, 6, 47, 187, 194, 202e, 212,

214, 215, 357, 450, 468e - molecular ion, 45-48, 4ge, 478-481 - molecule, 483-485

I icosahedron, 287 identical descriptions, 275 improper mixture, 263 incoherent radiation, 455 independence (linear), 12 infrared, 475, 489 instrumentalist, 114, 123 integra-differential equation, 195 interference, 3, 8, 150, 269 International Units, XV, 145 interpretation of formalism, 110 interpretation of state vector - epistemic, 112, 516 - individual, lll, 509 - statistical, lll, 516, 521 interpretation rule, 31-36, 73, 256, 271 invariance of Hamiltonian, 276 inverse scattering problem, 238 ion, 41, 183n, 198, 225, 413 - hydrogen-like, 6, 187, 212, 214, 357 ionic bond, 485 ionization, 384 - energy, 189n, 198, 432 - of hydrogen, 450-453 iron, 200-201

5.')4 Index

irreducibility, 59, 283 irreducible tensor operator, 327, 334e isolated quantum system, 31, 33, 256,

265, 509 isomorphism, 280

J Jacobi polynomial, 154 Jacobian, 371 jj coupling, 437 Josephson effect, 88 Jost function, 245

K kelvin, XV ket, 13, 101 kilogram, XV kinetic theory of gases, 2 Klein-Gordon equation, 341 Koopmans's theorem, 405 Kronecker delta, 17 krypton, 200-201

L laboratory frame, 227 Lagrange multiplier, 391, 402 Lagrangian, 365 Laguerre polynomial, 142, 155 Lamb shift, 358 A-doubling, 482n Lande factor, 64, 439 Laplacian, 139, 184 Larmor frequency, 66 Larmor precession, 66 laser, 455n LCAO method, 477 Legendre function (associated), 137,

152 Legendre polynomial, 151, 239 Levi-Civita symbol, 134 Lie algebra, 295, 306 Lie group, 292-296 lifetime, 465 light, 3 limit, 100 linear functional, 13, 101 - continuous, 101 linearity - of operator, 14

- of scalar product, 12 Lippmann-Schwinger equation, 231 liquid, 249 lithium, 194, 200-201, 432 local theory, 505 locality, 506 logic, 386, 521 - sensible, 522 logical implication, 270 logical structure, 270 Lorentz group, 337 - fundamental representation, 341 - Lie algebra, 340 - orthochronous, 338 - proper, 338, 339n - proper orthochronous, 339 - spinor representation, 343 Lorentz transformation, 338 - special, 341 LS coupling, 416, 438, 460

M magnesium, 200-201, 432 magnetic dipole transition, 464 magnetic field, 5, 54, 145, 291, 437 magnetic induction, XVI, 54, 145 magnetic moment, XVI, 53-55, 64, 147,

433, 464, 511 - of electron, 351 manganese, 200-201 Maple V, 22n, 97n, lOIn, 154-156n mass, 33 Mathematica, 22n, 97n, 102n, 154-156n mathematical formalism, 109 matrix, 18 - block-diagonal, 23 - diagonalization, 23, 25 - of adjoint operator, 19 - of inverse operator, 19 - of product, 19 - orthogonal, 23 matrix representation, 281-284, 307,

308, 310, 481 - of Lorentz group, 341, 343 Maxwell's equations, XVI, 3, 145 mean free path, 2 mean value, 35, 36, 256 - time variation, 45 measurement

- of dynamical variable, 33, 36, 115-118, 508, 521

- of spin, 115 - of type I, 510 metal, 5, 7 meter, XV metric tensor, 337 microcanonical distribution, 265 microobject, 110, 115, 116, 122, 258,

508 - identity, 111 microwaves, 475, 489 mixture, 257, 263 molecule, 1, 2, 31, 41, 53, 110, 225, 284,

471 - diatomic, 218, 481, 497e - polyatomic, 493 - stability, 473 moment of inertia, 494 momentum, 33, 118 - four-vector, 344 - operator, 73, 77, 128, 148, 327 momentum representation, 76 monoclinic, 299 multiplet, 416 - energy, 428-431 - wave functions, 417-422 multiplication table, 277

N N 2 ,490 natural shape, 465 neon, 195, 200-201, 432 Neumann function - spherical, 143, 157, 242 neutron, XV, 122, 225 Newton rings, 3 NH3 , 300e, 493 nickel, 200-201 nitrogen, 194, 200-201, 432 NO, 483 node of a lattice, 298 node of radial function, 139n norm - of discrete function, 169 - of ket I E), 79 - of ket Ip), 74 - of ket Ir), 127 - of ket Ix), 72

Index 555

of vector, 13, 32, 34, 41, 130, 140 normal mode of vibration, 6, 494 normal subgroup, 280 normalization, 13, 207 nuclear force, 384 nuclear radius, 183 nucleus, 5, 31, 110, 183, 211, 225, 384,

471 number of quantum states, 381 Numerov method, 166

o 0,287 O2 ,483,490 Oh, 287, 299 0(3), 306 0(3,1),338 O(n), 296, 301e O(p, q), 296 observable, 33 operational definition, 110 operator, 14 - adjoint, 15, 101 - annihilation, 90 - antilinear, 276n - anti unitary, 276n, 290 - commutation, 25, 44 - creation, 90 - domain of, 101 - exponentiation, 24, 293 - function of, 24 - Hermitian, 15, 33, 77, 101, 129 - identity, 15, 19, 277 - inverse, 15, 277 - linear, 14 - of permutation, 415 - self-adjoint, 101 - space inversion, 277 - symmetric, 399-400 - time-reversal, 291 - unitary, 15, 276 optical theorem, 242 orbital, 189, 194, 414, 474 - molecular, 476, 479 orbital angular momentum, 134-138,

148, 415 order of a group, 280 orthogonal complement, 61 orthogonality, 13, 72, 285

556 Index

orthorhombic, 299 oscillator, 7, 8 - damped, 519 overlap, 37 overlap integral, 491 oxygen, 195, 200-201, 430, 432

p pair annihilation, 346 pair creation, 346 parity, 277, 416n, 441, 460 partial trace, 262, 518 partial wave, 242 particle, 94 - elementary, 31, 53 - incoming, 225 - scattered, 225 partition function, 266 passive transformation, 20 path integral, 364 Pauli exclusion principle, 190, 289, 346,

396 Pauli matrices, 62 pendulum, 519 penetration factor, 383 periodic table, 195 permutation, 289n permutation group, 284, 289-290 perturbation, 206 - degenerate energy, 209-211 - harmonic, 448 - nondegenerate energy, 207-209 - time-dependent, 446 perturbed Hamiltonian, 206, 446 phase, 91 - of ket Ip), 75 - of ket IT), 297n phase shift, 178, 240, 241n, 243, 244,

253e phase space, 363, 381, 386 phenomena, 109 phosphorous, 200-201, 432 photoelectric effect, 7 photon, 7, 110, 121, 122, 225, 448, 505 - diffraction, 120 physical reality, 2, 499, 522 Planck constant, XV, 7, 39, 124 - universality, 121 Planck oscillator, 7, 8

plane wave, 344 planet, 1, 5 point group, 284, 286-287,474,493 Poisson brackets, 123 polarization, 3 polyatomic molecule, 493 position, 33, 118 - contravariant four-vector, 337 - covariant four-vector, 338 position operator, 46, 73, 127, 148, 327 positron, 346, 348n postulate, 9, 31, 33, 73, 133n potassium, 200-201 potential - barrier, 124, 382 - effective, 139, 384 - free particle, 83 - Gaussian, 172 - hard sphere, 243 - harmonic oscillator, 89 - r- 2 , 253e - r-1 , 140-143,330, 340n, 355-358 - r2, 329 - scattering, 227 - spherical square well, 143-144, 244 - spherically symmetric, 138, 173, 177,

189, 238, 354 - square barrier, 87-88, 105e - square well, 84-87, 105e - step, 105e, 133 - well, 124, 380 potential four-vector, 346 Poynting vector, 467e preparation (of state), 35 pressure, 2 probability, 112 - conditionnal, 264 - joint, 264 - of finding a particle, 80, 129 - of result, 34-36, 256, 521 - of tunneling, 382 probability current, 133, 133n, 15ge,

161e, 228, 252e, 347 product law, 277 projection postulate, 117, 513 projective representation, 309 projector, 15, 23, 36, 37, 101, 256, 267 - spectral decomposition, 23

propagator, 82, 107e, 361, 387e - of free particle, 93, 131, 369 proper mixture, 263 property, 37, 267 - classical, 385 - probability of, 267 proposition, 270 - reliable, 522 - true, 522 protein, 493 proton, XV, 45, 183

Q quadrupole moment, 328 - electric, 464 quanton, lIOn quantum number, 143 quantum of action, 122 quantum optics, 97 quantum states - number, 381 quantum system, 31, 33, 38 - isolated, 31, 33, 256, 265, 509 - total, 260, 319, 500 quasi-projector, 386n

R Racah formalism, 417 radial equation, 139 radial function, 139 - node, 139n radiation, 5 - intensity, 6, 455 realist, 114 reality, 499, 522 reduced mass, 5n, 185 reducibility, 59, 283 - complete, 283 reductionism, 508 reference frame, 337 - of nuclei, 474 reflection, 3, 4 reflection coefficient, 133 reflection in a plane, 286 refraction, 3 relative coordinates, 185 relativistic effect, 184, 353, 441 relativistic locality, 506 residue theorem, 233

Index 557

resolution of identity, 19, 72, 75, 80, 127, 130

resonance, 247 rigid body, 315, 318 Rodrigues's formula, 151 rotary reflection, 286 rotation, 56, 306, 341 - generator, 57, 59-62 - infinitesimal, 56 - matrix, 68, 310-314, 318-319, 325,

487, 495 - of molecule, 474 - proper, 306 rotation energy, 221, 475, 487, 488 rotation group, 278, 292, 306, 312 rotation operator, 278, 312 row matrix, 18 rubidium, 4 Runge-Lenz vector, 330n Rutherford model, 5 rydberg, XV, 188

S,82 S2, 299 S2n, 287 S3,300e Sn,289

s

scalar potential, XVI, 145 scalar product, 12, 32, 100, 259 scandium, 200-201 scattering - elastic, 226 scattering amplitude, 228, 234, 240 scattering center, 225 scattering integral equation, 234 scattering matrix, 242 Schrodinger equation, 38, 41, 81, 146,

260 - particle in three dimensions, 129 - solution, 42, 43, 82, 129, 130 Schrodinger picture, 41 Schur lemmas, 285 Schwartz inequality, 13, 27e second, XV, XVI secular equation, 22 selection rule, 287, 288, 460, 464,

488-493 selenium, 200-201

558 Index

self-adjoint operator, 101 self-consistent, 196, 478 self-consistent solution, 404 semiclassical approximation, 370 semi simple group, 280 separability, 100 shell, 189, 414 - closed (or filled), 196,416,428 SI units, XV, 145 silicium, 200-201, 430, 432 simple group, 280 simple locality, 506 singlet, 398, 502 singular matrix, 295 singularity, 233 - of Green's operator, 230 - of partial wave, 246 SL(2; R), 302e SL(n; C), 295 SL(n; R), 295 Slater determinant, 398, 414 SO(3), 301, 302e, 306, 329 SO(3) @ SO(3), 332 SO(3, 1), 338 SO(n), 296, 301e SO(p, q), 296, 342 sodium, 200-201, 432 solenoid, 150 space group, 299 space inversion, 277, 306, 339, 347,

416n, 472 specific heat, 7 spectral decomposition, 21, 33, 101 - of function of an operator, 24 - of Hamiltonian, 38 - of projector, 23 spectroscopic notation, 189, 416 spectrum, 4, 9, 16 - continuous, 79, 87, 130, 170, 177,

228, 449 - discrete, 79, 81, 86, 89, 130, 139, 142,

143, 167, 174, 206, 392, 449, 453 speed of light, XV, 8 spherical Bessel function, 143, 157, 240,

242 spherical coordinates, 136 spherical harmonic, 137, 153, 326

spherical Neumann function, 143, 157, 242

spherical square well, 143-144, 244 spin, 33, 53, 63, 64, 226, 301e, 415 spin 1/2, 66-69, 112, 190, 268,

272-274e, 333e, 343, 500, 502 spin 1, 70e spin 3/2, 70e, 114 spin-orbit interaction, 353, 433-437,

439, 488 spinor, 343 - chiral, 348 spontaneous localization, 515n square potential barrier, 87-88, 105e square well potential, 84-87, 105e standard deviation, 35, 36 Stark effect, 5, 215, 222e statcoulomb, XVI state - classical, 385 - excited, 47 - ground (fundamental), 47 state preparation, 35, 115-116, 258, 523 state space, 32, 33, 46, 55, 185, 445 - of particle, 72 - of particle in one dimension, 80 - of particle in three dimensions, 127 state vector, 32, 41, 42 state vector collapse, 117, 513 stationary perturbation, 433, 438 - degenerate energy, 435 stationary state, 42 stationary-phase approximation, 375 statistical mixture, 257 statistical operator, 256 statistical physics, 2, 112 step function, 100 step potential, 105e, 133 Stern-Gerlach device, 115, 511 Stern-Gerlach experiment, 53-56, 63,

64,66 strictly forbidden transition, 457 strong locality, 506 structure constants, 295, 303e SU(2), 308 SU(3), 330 SU(n), 295, 301e SU(p, q), 295

subgroup, 277, 279 subshell, 189n subspace, 12 ~ invariant, 59n subsystem, 261 sulfur, 7, 200~201, 430, 432 superconductor, 31 superfluid, 31 superluminal, 505 symbol ~ 3j, 324 ~ 6j, 326 symmetric molecular function, 493 symmetry group, 277 symmetry transformation, 276

T T,287 T d ,287 T h ,287 temperature, 2, 266 tensor, 341 ~ electromagnetic field, 349 tensor product, 259, 284, 296, 397 term, 436 ~ molecular, 482 tetragonal, 299 tetrahedron, 287 thermal conductivity, 2 thermal motion, 465 thermodynamics, 2, 6 Thomas~Fermi equation, 190, 192, 202e Thomas~Fermi model, 202e, 407 time reversal, 290~292, 301e, 339, 347 titanium, 200~201 top, 495 total angular momentum, 434, 472 total system, 260, 319 trace of an operator, 20 trajectory, 363, 365 ~ classical, 365, 384 transformation ~ active, 20, 37 ~ passive, 20 transition ~ electric dipole, 459, 488 ~ electric quadrupole, 464 ~ forbidden, 459 ~ fundamental, 490

~ magnetic dipole, 464 ~ rotational, 490 ~ strictly forbidden, 457 ~ vibration-rotation, 490

Index 559

translation group, 278, 292, 296~299 translation operator, 278 transmission coefficient, 134 transpose matrix, 18 triclinic, 299 trigonal, 299 triplet, 398 tritium, 187, 202e turning point, 377

u U(2), 302e U(n), 295, 301e ultraviolet, 7, 489 uncertainty principle, 119~ 122, 466 unit element, 279, 292 unitary representation, 281, 294, 298 units ~ Gaussian, XVI, 184n ~ 81, XV, 145 universes (distinct), 514 unperturbed Hamiltonian, 206, 446

V van der Waals forces, 215~218 vanadium, 200~201 variance, 35 variational method, 392 vector, 11 ~ addition, 11 ~ components, 12 ~ expansion, 12 ~ normalization, 13, 32, 34, 130, 140 ~ orthogonality, 13, 72 ~ scalar multiplication, 11 vector potential, XVI, 145, 149~ 151 vector space, 11, 32, 283 ~ complex, 11, 32 ~ dual, 13 ~ isomorphic, 32 - real, 11 ~ subspace, 12 vibration energy, 219~220, 475, 487,

488, 494 viscosity, 2

560 Index

visible light, 489 voltaic pile, 3

W wave function, 81, 128 - antisymmetric, 396-398 - boundary conditions, 129, 129n - normalization, 131 wave packet, 94-97, 124 wave-particle duality, 8, 122 Wigner distribution, 385n Wigner-Eckart theorem, 329, 435, 460

WKB approximation, 377, 388, 38ge

X-rays, 8

Y,287 Y h ,287

x

y

Z Zeeman effect, 5, 6, 65, 439 zinc, 200-201