Errata -Classical and Quantum Statistical Physics

November 23, 2024 Errata - Classical and Quantum Statistical Physics C. Heissenberg a and A. Sagnotti b a Nordita, Stockholm University and KTH Royal Institute of Technology, Hannes Alfvéns väg 12, 106 91 Stockholm, Sweden Department of Physics and Astronomy, Uppsala University, Box 516, 75120 Uppsala, Sweden e-mail: [email protected] b Scuola Normale Superiore and INFN Piazza dei Cavalieri, 7 56126 Pisa ITALY e-mail: [email protected] • p. 9 - In eq. (1.47) the energies in the derivatives should be U1 and U2 . • p. 10 - In eq. (1.48) the volumes in the derivatives should be V1 and V2 . • p. 12 - At the end of Section 1.6, “chemical grows” should be “chemical potential grows”. • p. 12 - N should not be present in the second log in eq. (1.73). • p. 18 - In eq. (2.6) the (2, 1) element should have a “+” sign. • p. 20 - In line 15 H should be H. • p. 21 - In eq. (2.29) the last two terms should have a factor kB . • p. 22 - Just above “Microcanonical Ensemble”, “the Planck’s constant” should be “Planck’s constant”. • p. 24 - P should be replaced by P T. • p. 24 - “surface and per unit time” should be “per unit surface and unit time”. • p. 25 - Eq. (2.59) should read ∂σ = − 1 − log pi − λ = 0 . ∂ pi • p. 25 - Eq. (2.61) should read 1 ∂2σ = − δij . ∂ pi ∂ p j pi • p. 27 - The normalized f (p) should be f (p) = 4 π  β 2πm 3 2 p2 p2 e−β 2m . • p. 29 - “Equations. (2.52)..” should be Eqs. (2.52) ..”. • p. 35 - In problem 2.2 d should be D. • p. 36 - In problem 2.4 d should be D. • p. 41 - In eq. (3.12) ⟨x| should read ⟨x|. • p. 42 - In eq. (3.32) m is the exponent should be m2 . • p. 47 - Eq. (3.32) should read ψλ (x) = m ω 1 4 πℏ 1 e − 2 |λ| 2 − 1 2 λ2 + 2λx √ mω 2ℏ − mω 2ℏ x2 . • p. 52 - r should be x near the top. • p. 65 - Bottom of the page: the inversion point should be x⋆ , for consistency with the notation in the following. • p. 71 - Equal signs are missing in the first column of Eq. (3.237) for the values of g(s). • p. 75 - The argument of the sine function should be π ∆n ℏω . • p. 78 - The prefactor of the integral in eq. (3.311) should be m ω η2 2 . • p. 79 - In eq. (3.313) the measure should be [D x(τ )]. • p. 79 - In eq. (3.317) the measure should be [D ξ(τ )]. • p. 80 - y0 = yN +1 = 0 should read ξ0 = ξN +1 = 0. • p. 98 - In eq. (4.6) the factor N kB should be 3 N kB . • p. 98 - Eq. (4.6) should contain the additional constant term − N log g0 . • p. 99 - In eqs. (4.17) and (4.18) V should replaced by V T . • p. 105 - would eventually lie at the heart of laser systems → that would eventually lie at the heart of laser systems. • p. 106 - dx is missing in the integrand. • p. 108 - Better to say “to first order in β”, which makes the selected contribution unique. 2 • p. 112 - The sentence “The Lamb shift (is discussed in Section 5.2) a key example in this respect ..” should read “The Lamb shift (discussed in Section 5.2) is a key example in this respect ..” • p. 113 - In eq. (5.3) the sum should start from n = 1. • p. 116 - In eq. (5.29) αe should be α. • p. 119 - In eq. (5.53) G should be GN . • p. 134 - “the Chapter 6” should read “Chapter 6” • p. 134 - Eq. (7.5) should read 1 ∂2σ = − δI I δN N . ∂ pI1 ,N1 ∂ pI2 ,N2 pI1 ,N1 1 2 1 2 • p. 138 - In eqs. (7.31) and (7.32),  2πm h2 β  should be  2πm h2 β 3 2 . • p. 144 - In eqs. (8.25) and (8.26) the combinatorial factors should be accompanied a • p. 144 - In eq. (8.29) nα gα nα ! should be replaced by Q nα gα α nα ! . Q α. • p. 152 - and the following. The Fermi energy is sometimes called EF and sometimes ϵF . It should be EF everywhere. • p. 153 - In eq. (8.94) V should be N , the number of particles. • p. 156 - The first integral in eq. (8.109) should have an overall “−” sign. • p. 158 - One should separate the l = 0 term in eq. (8.123), writing N = G(µ) + 2 ∞ X l=1   G(2l) (µ)(kB T )2l 1 − 21−2l ζ(2l) • p. 158 - In the second line of eq. (8.121) the factor of g(ϵF ) should be (µ − ϵF ). One should similarly separate the l = 0 term in eq. (8.123). • p. 158 - One should separate the l = 0 term in eq. (8.120), writing U (t) = H(µ) + 2 ∞ X l=1   H (2l) (µ)(kB T )2l 1 − 21−2l ζ(2l) • p. 162 -Eq. (8.141) is correct but it would be clearer if T A were written in the opposite order: A is the area of the 2D system. • p. 168 - The factor 1 N should not be present in Eq. (8.181). 3 • p. 168 - The sentence related to the nuclear spin I should read: “The nuclear spin I is another source of degeneracy. It gives rise to the hyperfine splitting of electron levels, whose magnitude becomes comparable with kB T only at temperatures of about 0.1 K. Therefore, at room temperatures nuclear spins simply contribute degeneracy factors (2I + 1) to atomic ground states.” • p. 173 - In the whole page “!” should read “ω”. • p. 180 - It would be clearer if ϵ were replaced by −|ϵ| in eqs. (8.266) and (8.267). • p. 184 - Add at the end: More generally one can obtain two conditions, demanding continuity of U and its first derivative at the end of the charge distribution. The former determines ϵF , and there are more solutions for charge distributions terminating at a finite value of r. • p. 185 - In eq. (8.300) g(x) should read g(xF ). • p. 190 - In the equation “per” should be replaced by “for”. • p. 191 - In Ex. 8.17 we mean spin- 12 fermions and spin-1 bosons. • p. 191 - In Ex. 8.19 it would be better to replace ϵ by −|ϵ|, keeping the same notation as in the main body of the chapter. • p. 193 - M in Eq. (9.3) should be M. • p. 198 - In eq. (9.54) there should be no forbidden region. 1 2, since there is no transition to a classically • p. 200 - The “,” at the end of the first line should be removed. • p. 201 - The comment after eq. (9.193) should be amended. The second contribution is independent of B and does not contribute to the susceptibility, when spelled out in detail. The starting point is        X  1 EF X 1 1 θ EF − 2µB B n + +2 . θ EF − 2µB B n + γ n+ M = −8 µB 2 2 B0 n 2 n to be considered for N B N < < , 2 (j + 2) B0 2 (j + 1) so that the first j + 1 levels are full and the j + 2-nd level is only partly full.
Recalling that EF = 2µB B j + 32 and γ = BB0 , the sums become " j   # X  3 1 + (N − γ (j + 1)) j + γ n+ M = −8 µB 2 2 n=0 " #   j X 3 + 4 µB j + γ 1 + (N − γ (j + 1)) . 2 n=0 4 As a result M = −8 µB      3 3 γ 2 (j + 1) + (N − γ (j + 1)) j + + 4 µB N j + , 2 2 2 and collecting the different terms finally gives M = 4 µB γ (j + 1) (j + 2) − 4 µB N so that χ =  3 j+ 2  , 4 µB (j + 1) (j + 2) > 0 . B0 • p. 201 - In eq. (9.76) B should be Bz . • p. 201 - Eq. (9.82) should read M = − 4 µB γ j (j + 1) . • p. 202 - In eq. (9.86) νB should be µB . • p. 206 - The last problem would be better formulated like this: ... with magnetic moments (±µ, ± 2 µ, ± 3 µ). • p. 215 - The product should not be present in eq. (10.60). • p. 218 - ξ ⋆ should be replaced with x⋆ in eq. (10.81) and in the following line. • p. 219 - Eq. (10.84) should read M = 1 ∂ log Z = N µ tanh(ξ ⋆ ) . β ∂B • p. 219 - The factor β should not appear in the second term in eq. (10.85). • p. 225 - The first expression in eq. (10.133) should read 2D a2 βv (βv − 1). • p. 226 - Eq. (10.146) should read χ ∼ |t|− γ . • p. 230 - After eq. (10.177), the text should read C = 1 2 Tc for T < Tc . • p. 233 - In eq. (10.193) the first coefficient should read − 12 . • p. 233 - The Helmoltz free energy should be denoted by A, not by F , as elsewhere. • p. 236 - In Ex. 10.5 the σi should be ni for uniformity of notation. • p. 239 - The beginning of Ex. 10.13 should read ”Consider a one–dimensional spin chain ...” 5 • p. 252 - One should stress that the directions of entry and exit coincide since the contributions concern the same values of p and q, which justifies ending up with a trace. • p. 253 - The factor N should not be present in Eqs. (11.49) and (11.50). • p. 309 - Eq. (13.217) should read m0 =  Z2 Z1 1 2 m • p. 318 - After Eq. (14.30) “lower end” should read “upper end”. • p. 319 - In eq. (14.41) P should read f . • p. 322 - Eq. (14.52) should read log f = α − m (v − u)2 2 kB T . • p. 330 - after eq. (14.108) remove ”which”. • p. 337 - In eq. (A.13), last line, ” + −” should read ” − ”. • p. 334 - The whole discussion is confused and not to the point. The reader is kindly asked to ignore it, starting from line 2 and to resume reading when she/he gets to the last paragraph. We cannot track how such incorrect statements slipped in, but for one matter 2 2 two-term recursion relations would obtain factoring out e±z /4 , i.e. letting ψ = e±z /4 χ. We are grateful to M. Barbieri for calling this issue, and several misprints, to our attention. • p. 338 - In the last of Eqs. (A.23) the integrand should be (x − x0 )2 f (x). • p. 348 - No need to move to the next line before “Eq. (D.1) by ...”. • p. 349 - In Eq. (D.11) the series should start from m = 0. • p. 356 - The sign of the σ · B in eq. (F.9) should be +. 6