Errata for Quantum Mechanics by Ernest Abers August 1, 2004

Transcription

1 Errata for Quantum Mechanics by Ernest Abers August, 004 Preface Page xv: Add a sentence at the end of the next-to-last paragraph: Rudaz (University of Minnesota, and several anonymous reviewers. My special thanks to Kenneth Lane and his students at Boston University. I thank the Department of Physics and Astronomy at UCLA for granting me Chapter I Page 4, Equation (.78: The last expression needs an ɛ in front: G = p m,i δr m,i = ɛ ɛ ijk p m,i n j r m,k = ɛ m,i ijk m (r m p m ˆn = ɛl ˆn (.78 Chapter II Page, Equation (.: In the leftmost expression, change to h: h p = λ = π me. Å (. Page 5, Equation (.6a: Change the last α to β: [ ] α + β ψ = α ψ + β ψ (.6a Chapter III Page 75, Equation (3.80: Put an in front of the left-hand side: [r i,l j ]=i k ɛ ijk r k (3.80 Page 84, Equation (3.37: The last term should be Ze /r: H = p m Ze r (3.37 Page 90, Equation (3.7. Change m to M. n l =M (3.7

2 Quantum Mechanics Chapter IV Page, just above Equation (4.05, insert solutions between the and to : The eigenstates are linear combinations of, / and 0, /. The eigenvalues are the solutions to the characteristic equation: Page 6, third paragraph, third line: change /f to /: Unless l = 0, there are two states with m = l /. One linear combination will be in the j = l +/ ladder. The remaining one must be the top of a new ladder, this time with j = l /. The number of states in these two ladders adds Page 3, Problem 4., part (b: Change the superscript j to : U am D ( ( J i mn U nb = ( J i mn Page 3, Problem 4.4, part (c. The last equation on the page should be changed to ab ψ(r =R ± El (rym l ± (θ, φ Chapter V Page 40, line below Equation (5.6: Change g(a tog(ɛ: For a one-parameter subgroup of continuous symmetries g(ɛ = exp( iɛg, Page 6, Equation (5.3: Divide the exponent by : U(t f,t i =e ih(tf ti/ (5.3 Chapter VII Page 03, Equation (7.8: In the first line change E n o to Eo n (twice: (H E o n + E o n H o ψ n = H ψ n (H o E o n ψ n =( n H ψ n (7.8 Page 05, second bulleted paragraph, seventh line: Change so to but : Two states in one of these ladders will have the same unperturbed energy, but we Page 4, Equations (7.69 to (7.70: Change the subscript 3 to D : H D = 8m V (r (7.69 E D = α4 m n 3 δ l0 (7.70

3 ERRATA 3 Page 6. Equation (7.33: Delete the subscript i (four times: ( p ψ 00 (r m σe ψ 00 (r d 3 r = E o r (σ =σ E o (7.33 Chapter VIII Page 70, Equation (8.40: change the left-hand side to f ( (Θ: f ( (Θ = e iq r U(r d 3 r 4π = r dr e iqr cos θ U(rd cos θ = r sin(qru(rdr 0 q 0 (8.40 Page 70, Equation (8.39, replace r by r inside the integral. f ( (θ, φ = e i(k k r U(r d 3 r (8.39 4π Page 73, Equation (8.56: In the second line, second term inside the large parentheses, move the asterisk from χ to φ. In the fourth line, first expression, second term inside the parentheses, change plus to minus: ( kσ tot (π = Im φ ( χ 3 r +χ ( φ r r dω ( = Im φ ( χ r χ( φ r r dω ( = Im φ ( ψ r ψ( φ r r dω (φ = Im (φ ψ ψ φ ˆndS = Im ψ ψ φ d 3 r = Im φ (ru(rψ(rd 3 r = Im f(0 (8.56 π Page 77, Equation (8.80, in the first line, in the numerator of first factor, change l + l to l +: A l ρ l l l! l + = (l +! n = i l ρ l l (l! l! (l! i n ρn n (n! P n (zp l (zdz n! (n! (8.80 Page 80, Equation (8.97: Change the last r to r. Move the asterisk to after the second Yl m factor.

4 4 Quantum Mechanics Page 80, Equation (8.98 Change the arguments of the first Yl m to (θ,φ. Page 80, Equation (8.99 second line, change ˆn to θ,φ, and change ˆn z to 0,φ: φ k (r = 4π i l Y (π 3 l m (θ, φyl m (θ,φ j l (kr (8.97 l,m and ψ k (r = 4π i l Y (π 3 l m (θ,φ Yl m (0,φR l (r (8.98 l,m Therefore the scattering amplitude is f(θ, φ = 4π U(r = 0 ( l l,m ( i l Y m l (θ, φ Yl m (θ,φ j l (kr i l Yl o (θ,φ R l (r Yl o (0,φ r dr dω [ ] (l +j l (kr U(r R l (r P l (cos θ r dr l (8.99 Page 80, Equation (8.00 On the left, k should be the denominator of all the rest, not just δ l : e iδl sin δ l = j l (kru(rr l (rr dr (8.00 k 0 page 80, last sentence, replace in that number by in that limit. Page 84, Problem 8., in the line just following the unnumbered displayed equation, on the right-hand-side of the in-line equation, remove the middle vertical line: where ρ(r = ψ00 (r, the probability distribution of the bound electron. Page 87, Reference [3], Replace the comma after K by a period: [3] K. Gottfried, Quantum Mechanics, Volume I, Benjamin, 966. See also Appendix D. Chapter IX Page 88, in the subsubsection heading just below the 9.. subsection heading, change Transistions to Transitions. Page 89, Equation (9.0, second line, 4m in denominator should be just m. P b (t V ba e i(ωba ωt ω ba ω (9.0 = α m A o ψ b p ˆεe ik r ψa sin( ω ba ωt/ ω ba ω

5 ERRATA 5 Page 90, Equation (9.6, replace 4m by m and 8π by 4π. P (t t α 4π m ρ(ω ba ψ b p ˆεe ik r ψn t (9.6 Page 9, Equation (9.7, replace 4m by m and 8π by 4π. ω ba Γ b = α m 4π ω ba ρ(ω ba ψ b p ˆεe ik r ψ n (9.7 Page 9, in the line just above Equation (9. change θ to cos θ. In Equation (9. change cos θ to cos θ: where θ is the angle between the constant vector r and the polarization. If the radiation is unpolarized and isotropic, just replace cos θ by its average value: cos θ cos θdω= (9. 4π 3 Page 9, change the text between Equations (9.5 and (9.6 to read: Then a ba (ω is related to A ba (ω by Page 94: There is no Equation (3.33. Latex just skipped an equation number! I don t really understand how this could have happened, but it is under control. Page 94, in the text just below Equation (9.30, change has an inverse to exists : Even though the operator H ω has no inverse for positive real ω, G(ω exists for real ω and any finite real ɛ. Page 95, in the sentence above equation (9.4, change the first (9.40 to (9.39: From equations (9.39 and (9.40 it also follows that Page 30, just before the Optical Theorem heading, add The relativistically correct forms of Equations (9.7 and (9.7 are obtained replacing m by ω a. Page 304, Equation (9.90: Change the left hand side to φ b T φ a : φ b T φ a = δ 3 (K K φ b T φ a (9.90 Page 304, Equation (9.93: The integrals should end in d 3 k d 3 k and d 3 K d 3 k : Γ =π T ba δ (ω a ω b d 3 k d 3 k δ3 =π (K K T ba δ (ω a ω b d 3 K d 3 k π T ba δ (ω a ω b δ 3 (K V K (π 3 d3 K d 3 k (9.93

6 6 Quantum Mechanics Pages , In equations (9.90 through (9.95 change T to T everywhere. Page 306, Equation (9.03. In the first line exchange r and r in the exponent. In the second line change the sign of the exponent: H ba, = ± V ( r (π 6 r e i(k k r e i(k k r d 3 r d 3 r = ± (9.03 V (re i(k+k r δ (π 3 3 (K K d 3 r Page 307: In equation (9.04 delete the delta function δ 3 (K K in the first line. Change T to T everywhere in (9.04 and (9.05. And replace p by k in these and the next two equations: T direct (ω a,θ,φ= (π 3 V (re i(k k r d 3 r = e π q = e (9.04 4π k ( cos θ = α 8π k sin (θ/ T exchange (ω a,θ,φ= T direct (ω a,π θ, φ +π dσ dω =(π4 µ α 64π 4 k 4 = α µ 4k 4 ( e = 6π k cos (θ/ = α (9.05 8π k cos (θ/ sin (θ/ ± cos (θ/ = α µ 4k 4 sin (θ/ ± cos (θ/ sin (θ/ cos (θ/ (9.06 sin 4 (θ/ + cos 4 (θ/ sin (θ/ cos (9.07 (θ/ cos 4 (θ/ + sin 4 (θ/ ± ( dσ = α µ ( dω unpolarized 4k 4 Page 3, Equation (9.3: On the left, change ω to ω a. On the right, change the denominator to ω b ω a. In Equation (9.33 replace ω by ω b. In the text below Equation (9.34 delete with ω a =Reω above : ω a = ω a + H aa +Re b a π b a H ba ω b ω a + (9.3 H ba δ (ω b ω a = Γ/ (9.33 G(ω aa = ω ω a + iγ/ Evidently T (ω has a singularity in the lower half plane, (9.34

7 ERRATA 7 Page 34 just below Equation (9.4 change closer to to farther from : which is farther from unity than the same probability Page 35, Equation (9.48 change absh ba to H ba : d dt a ba(t H ba (ω a ω b sin(ω a ω b t (ω a ω b +Γ /4 (9.48 Chapter X Page 340, Figure 0.. The arcs should form a circle, whose center is at the base of the arrow. [Some glitch is a graphics program printed some of the arcs in the wrong place.] page 355, Reference [4], change Tycho to Tycko, here and in the index. [4] R. Tycko, Phys. Rev. Letters 58 ( Chapter XI Page 365, Equations (.57 and (.57: In both equations change 4pi to 4π. E(r,t= A t = (4π/ (π i ω [ a(k,t a ( k,t ] e ik r d 3 k (.57 3/ ω and B(r,t= A = (4π/ (π 3/ i ω ωˆn [ a( k,t+a (k,t ] e ik r d 3 k (.58 Page 377, Equation (.9. Replace 5 by 6, and therefore replace by and replace by The the next line needs to read The lifetime is seconds, over 00,000 centuries.... Also, delete Equation (.7e there is nothing there: The J = 0 state is,, M = 0 (.7d Γ= α3 gp(m/m 3 3 m = ev = sec (.09 The lifetime is seconds, over 00,000 centuries. It should be clear Page 403 Problem.5: Delete the sentence What is the width of these components of the Lyman-β line? Compute the rate for the spontaneous decay of a 3P state to the S state. you can ignore spin these are electric transitions. (Here

8 8 Quantum Mechanics Chapter XII Page 409, Equation (.4c: Change x to z: z = z (.4c Page 40, Equations (.0: The exponents should be boldface: R = e iθˆn J B = e iω ˆn K (.0a (.0b Page 46, Equation (.67: Change the sign of the last term inside the large parentheses from minus to plus: ( d l(l + α m dr mr + ωα mr + ω m u(r = 0 (.67 m Chapter XIII Page 439, Equations (3.9 through (3.: Change pi to π four times: Ψ(r = e ip r a(pd 3 p (3.9 (π 3/ Then Ψ (r = (π 3/ e ip r a (p d 3 p = (π 3/ e ip r p d 3 p (3.0 This is a one-particle state whose wave function is ψ(r = r Ψ (r = e ip r r p d 3 p = δ (π 3/ 3 (r r (3.