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Believed to be typos in Desai’s QM Text

Posted by peeterjoot on June 19, 2011

[Click here for a PDF of this post with nicer formatting (especially if my latex to wordpress script has left FORMULA DOES NOT PARSE errors.)]

This is a refresh of the earlier typo notes posted previously.

I have found some of the obvious stuff in my reading of \citep{desai2009quantum}. Prof.\ Vatche Deyirmenjian who teaches our PHY356 course has pointed out still more (and pointed out where I’d identified the wrong source for some typos).

Chapter 1.

\begin{itemize}
\item Page 1. Prof.\ Deyirmenjian: The Hermitian, not complex conjugate, of {\lvert {} \rangle} is {\langle {} \rvert}.
\item Page 5-6. Prof.\ Deyirmenjian: Change the {*} in (1.26), (1.31), and (1.33) to a dagger.
\item Page 7. Text before (1.43). \alpha instead of a used.
\item Page 19. Equation (1.122). \daggers omitted after first equality.
\end{itemize}

Chapter 2.

\begin{itemize}
\item Page 40. Text before (2.137). Reference to equation (2.133) should be (2.135)
\item Page 53. Is the “Also show that” here correct? I get a different answer.
\end{itemize}

Chapter 3.

\begin{itemize}
\item Page 61. Equation (3.51). 1/\hbar missing.
\item Page 62. Equation (3.58). Prof.\ Deyirmenjian: Remove the U_I operators from Eq. (3.58)
\item Page 66. Equation (3.92). -(d/dt {\langle {\alpha} \rvert}) {\lvert {\alpha} \rangle} should be +{\lvert {\alpha} \rangle} d/dt {\langle {\alpha} \rvert}.
\item Page 66. Equation (3.93). H on wrong side of {\langle {\alpha} \rvert}
\item Page 74,76. Prof.\ Deyirmenjian: remove the extra brackets from Eq (4.9) and (4.21).
\item Page 79. Prof.\ Deyirmenjian: “The probability of finding this particle” should read “The probability density for this state at point x is”
\end{itemize}

Chapter 4.

\begin{itemize}
\item Page 81. Equation (4.52). Should be -2\alpha in the exponent.
\item Page 82. Equation (4.65). Prof.\ Deyirmenjian: a 1/\sqrt{2\pi} is missing before the integral. Note that without this (4.67) appears incorrect (off by a factor of \sqrt{2\pi}, but the error is really just in (4.65).
\item Page 82. Equation (4.67). Prof.\ Deyirmenjian: the negative sign should appear inside the large square brackets.

\item Page 83. Equation (4.74). A normalized wave function isn’t required for the discussion, but if that was intended, a 1/\sqrt{2\pi} factor is missing.
\item Page 83-84. Prof.\ Deyirmenjian: In (4.67) and (4.77), the derivative should be evaluated at k=k_0.
\item Page 86. Equation (4.99). Extra brace in the exponent.
\item Page 87. Equation (4.106). Extra brace in the exponent.
\item Page 89. Equation (4.124-4.130). Prof.\ Deyirmenjian: C e^{\pm \sqrt{\mu}\phi} is not a solution to (4.122). This should be Q(\phi) = C e^{i \sqrt{\mu} \phi} and (4.126) should be \sqrt{\mu} = m. This fixes the apparent error in sign in equations 4.129 and 4.130 which are correct as is.
\item Page 92. Equation (4.158). Prof.\ Deyirmenjian: should read P_l(1) = 1.
\item Page 93. Equation (4.169). conjugation missing for Y_{lm}. Y_{l'm'} is missing prime on the l index.
\item Page 95. Second line of text. Language choice? “We now implement”. perhaps utilize would be better?
\item Page 95. Text before (4.193). i is in bold.
\item Page 96. Text before (4.196). i is in bold.
\item Page 97. (4.205). i is in bold.
\item Page 97. (4.207-209). \mathbf{i}, and \mathbf{j}s aren’t in bold like \mathbf{k}
\item Page 101. (4.245). The right side should read Y_{l,m+1}
\item Page 101. (4.239-240). The approach here is unclear. FIXME: incorporate lecture notes from class that did this using braket notation.
\item Page 102. (4.248-249). Commas missing to separate l, and m\pm 1 in the kets.
\end{itemize}

Chapter 5.

\begin{itemize}
\item Page 109. (5.49). Remove bold font in right hand side state {\lvert {\chi_{n+}} \rangle}.
\item Page 113. (5.86). One \sigma isn’t in bold.
\item Page 114. (5.100). \chi is in bold.
\item Page 115. Text before (5.106). \alpha in bold.
\item Page 118. Switch of notation in problem 5 for ensemble averages. [S_i] used instead of \left\langle{{S_i}}\right\rangle_{\text{av}}.
\end{itemize}

Chapter 6.

\begin{itemize}
\item Page 120. \phi in bold. A not in bold.
\item Page 123. (6.26). 1/i \hbar factor missing on RHS.
\item Page 124. Text before (6.37). You say canonical momenta P_k, but call these mechanical momenta on prev page.
\item Page 125. (6.41). Some \psis are in bold.
\item Page 126. (6.49). There’s no mention that \mathbf{B} is constant, leaving it unclear how the gauge condition and how the curl of \mathbf{A} reproduces \mathbf{B}. This would also help clarify how you are able to write \boldsymbol{\mu} \cdot \mathbf{B} = \mathbf{B} \cdot \boldsymbol{\mu}.
\item Page 128. (6.65). \boldsymbol{\mu} \cdot \mathbf{L} should be \boldsymbol{\mu} \cdot \mathbf{B}.
\item Page 129. (6.75). \boldsymbol{\mu} \cdot \mathbf{L} should be \boldsymbol{\mu} \cdot \mathbf{B}.
\item Page 130. (6.80). integral looks like it should be \int_{\mathbf{r}' = \mathbf{r}_0}^\mathbf{r} \frac{e}{c \hbar} \mathbf{A}(\mathbf{r}') \cdot d\mathbf{r}'. ie: Clarify bounds, and add a factor of c in the denominator which is required for the cancellation of (6.82).
\item Page 131. (6.81,6.86). Factors of cs should be with each of the \hbars.
\item Page 131. Problem 1. bold missing on \mathbf{E}.
\end{itemize}

Chapter 8.

\begin{itemize}
\item Page 143. (8.58). \beta should be negated.
\item Page 159. (8.6.3). Two references to Chapter 2 should be Chapter 4.
\item Page 160. (8.199). Want \hbar^2 not \hbar in expression for k.
\item Page 162. (Fig 8.9). Figure is backwards compared to text (a bump instead of a well).
\item Page 165. (8.235). Extra R_l factor inside parens.
\end{itemize}

Chapter 9.

\begin{itemize}
\item Page 174. (9.5). Have \hbar/2m\omega instead of \hbar m \omega/2 in expression for P.
\item Page 181. (9.57). Factor of two missing. Want \frac{\alpha}{2 \sqrt{\pi}}.
\item Page 186. (Problem 10). Sequencing the text and problems is off. The green’s function technique isn’t introduced until chapter 10.
\end{itemize}

Chapter 10.

\begin{itemize}
\item Page 189. (10.22). It would be nice to have a reference to the appendix (ie: 10.100) for the chapter so that this identity isn’t pulled out of a magic hat.
\item Page 192. (10.44, 10.45). 2 \alpha {\alpha^{*}}' should be \alpha {\alpha^{*}}' + \alpha' \alpha^{*}
\item Page 193. (10.51). Application (slowly, step by step explicitly) of 10.100 to expand the e^{\frac{i}{\hbar}(p_0 X - x_0 P)} in the braket gives

\begin{aligned}{\langle {x} \rvert} e^{\frac{i}{\hbar}(p_0 X - x_0 P)} {\lvert {0} \rangle}&={\langle {x} \rvert} e^{\frac{i}{\hbar}p_0 X }e^{-\frac{i}{\hbar}x_0 P}e^{-\frac{i}{2\hbar}x_0 p_0 \left[{X},{P}\right]}{\lvert {0} \rangle} \\ &={\langle {x} \rvert} e^{\frac{i}{\hbar}p_0 X }e^{-\frac{i}{\hbar}x_0 P}e^{\frac{x_0 p_0}{2} }{\lvert {0} \rangle} \\ &=e^{\frac{x_0 p_0}{2} }{\langle {x} \rvert} e^{\frac{i}{\hbar}p_0 X} e^{-\frac{i}{\hbar}x_0 P}{\lvert {0} \rangle} \\ &=e^{\frac{x_0 p_0}{2} }\left({\langle {0} \rvert} e^{\frac{i}{\hbar}x_0 P}e^{-\frac{i}{\hbar}p_0 X} {\lvert {x} \rangle}\right)^{*} \\ &=e^{\frac{x_0 p_0}{2} }\left({\langle {0} \rvert} e^{\frac{i}{\hbar}x_0 P}{\lvert {x} \rangle}e^{-\frac{i}{\hbar}p_0 x} \right)^{*} \\ &=e^{\frac{x_0 p_0}{2} } e^{\frac{i}{\hbar}p_0 x} \left({\langle {0} \rvert} e^{\frac{i}{\hbar}x_0 P}{\lvert {x} \rangle}\right)^{*} \\ &=e^{\frac{x_0 p_0}{2} } e^{\frac{i}{\hbar}p_0 x} \left(\left\langle{{0}} \vert {{x - x_0}}\right\rangle\right)^{*} \\ &=e^{\frac{x_0 p_0}{2} } e^{\frac{i}{\hbar}p_0 x} \left\langle{{x - x_0}} \vert {{0}}\right\rangle \\ &=e^{\frac{x_0 p_0}{2} } e^{\frac{i}{\hbar}p_0 x} \psi_0(x - x_0, 0)\end{aligned}

This is the same as (10.51) with the exception of a real scalar constant e^{ x_0 p_0/2} multiplying the wave function. Because of this I think that (10.51) should be a proportionality statement, and not an equality as in

\begin{aligned}{\langle {x} \rvert} e^{\frac{i}{\hbar}(p_0 X - x_0 P)} {\lvert {0} \rangle} \propto e^{\frac{i}{\hbar}p_0 x} \psi_0(x - x_0, 0)\end{aligned}

(ie: building this additional factor into the wave function normalization instead).
\item Page 196. (text after 10.76). Looks like reference to Chapter 9, should be Chapter 9 problem 5.

\item Page 197. (text after 10.85). Reference to Chapter 1 should be Chapter 2.
\end{itemize}

Chapter 26.

\begin{itemize}
\item Page 486. (26.60). \mathbf{n} \times \mathbf{r} \cdot \boldsymbol{\nabla} ought to have braces and read (\mathbf{n} \times \mathbf{r}) \cdot \boldsymbol{\nabla}.
\item Page 496. (26.154). Remove Y_{l'm}(\theta, \phi) term from the integral.
\end{itemize}

Chapter 31.

\begin{itemize}
\item Page 562. (31.56). T' = L T \tilde{M} is given for a mixed tensor representation. This is T^\mu_{.\nu}. The other mixed representation T_{\mu}^{.\nu} transforms as T' = M T \tilde{L}.
\end{itemize}

Chapter 32.

\begin{itemize}
\item Page 575. minor: E t - \mathbf{p} . \mathbf{r} written instead of E t - \mathbf{p} \cdot \mathbf{r}
\item Page 576. minor: \boldsymbol{\nabla} . \mathbf{j} instead of \boldsymbol{\nabla} \cdot \mathbf{j}
\item Page 577. (32.23). \mathbf{j} accidentally includes the divergence.
\item Page 579. (32.35). Sign missing in exponential. Should be e^{-i k \cdot x} not e^{i k \cdot x}.
\item Page 579. \hbar \omega_k is the energy of the particle, not \omega_k. There’s also an \hbar missing in the expression for \omega_k. That is \omega_k = \sqrt{ c^2 \mathbf{k}^2 + m_0^2 c^4/\hbar^2}.
\item Page 580. (32.40). The factor of g presumed constant ought to be incorporated into \chi if this is to be consistent with the (32.45) that follows.
\item Page 583. (32.70). sign error. negate integral.
\item Page 584. (32.74). sign error in the both the square root and subsequent approximation, which should be p_4 = \pm \sqrt{ \mathbf{p}^2 + m^2 - i \epsilon} \approx \pm (\omega_p - i \epsilon'). (I’ve added approximately equal for the second part since that wasn’t specified which I found confusing).
\item Page 584. (32.75-76). there are multiple sign errors in these equations which should be

\begin{aligned}\frac{1}{{p^2 - m^2 + i\epsilon}} &\approx \frac{1}{{(p_4 - \omega_p + i\epsilon')(p_4 + \omega_p - i\epsilon')}} \\ &\approx\frac{1}{{2 \omega_p}}\left(\frac{1}{{p_4 - \omega_p + i\epsilon'}}-\frac{1}{{p_4 + \omega_p - i\epsilon'}}\right)\end{aligned}

Note that an attempt to confirm (32.76) yields

\begin{aligned}\frac{1}{{p_4 - \omega_p + i\epsilon'}}-\frac{1}{{p_4 + \omega_p - i\epsilon'}}=\frac{2 \omega_p - 2 i \epsilon'}{ p^2 - m^2 + i \epsilon + \epsilon^2/4(\mathbf{p}^2 + m^2)}\end{aligned}

So we need approximations twice for the “equality”.

\item Page 584. (before 32.78). minor: bold script used for \mathbf{p} \cdot \mathbf{r} on second like of the change of variables.
\item Page 585. (32.82, 32.83). minor: p_n . x instead of p_n \cdot x.
\item Page 585. (32.82, 32.83). wrong normalization? wouldn’t we want 1/\sqrt{2 \omega_{p_n}}.
\item Page 585. (32.84). notation switch? 0n index whereas n0 used above? What are the definitions of \psi_{0n}? that allow the integral to be converted to a sum?
\item Page 586. (32.88). minor: i \mathbf{k} . instead of i \mathbf{k} \cdot
\item Page 586. (32.88). I calculate a negated G(\mathbf{k}) from (32.88). Guessing that (32.88), and (32.93 on pg 587) where intended to be negated like done earlier (for example in (32.57)).
\item Page 587. (32.93). minor: i \mathbf{k} . instead of i \mathbf{k} \cdot
\item Page 588-589. (32.104-105). This substitution doesn’t appear to work?

\begin{aligned}&(\boldsymbol{\nabla}^2 - \mu^2)(\phi' e^{-\mu r}) \\ &= \frac{1}{{r^2}} \frac{\partial {}}{\partial {r}} \left( r^2 \frac{\partial {}}{\partial {r}} \left( \phi' e^{-\mu r} \right) \right) - \mu^2 \phi' e^{-\mu r} \\ &= \frac{1}{{r^2}} \frac{\partial {}}{\partial {r}} \left( r^2 \left( \frac{\partial {\phi'}}{\partial {r}} e^{-\mu r} -\mu \phi' e^{-\mu r} \right) \right) - \mu^2 \phi' e^{-\mu r} \\ &= \frac{\partial {}}{\partial {r}} \left( \left( \frac{\partial {\phi'}}{\partial {r}} e^{-\mu r} -\mu \phi' e^{-\mu r} \right) \right) + 2 \frac{1}{{r}} \left( \frac{\partial {\phi'}}{\partial {r}} e^{-\mu r} -\mu \phi' e^{-\mu r} \right)  - \mu^2 \phi' e^{-\mu r} \\ &= \frac{\partial^2 {{\phi'}}}{\partial {{r}}^2} e^{-\mu r} -\mu \frac{\partial {\phi'}}{\partial {r}} e^{-\mu r} -\mu \left( \left( \frac{\partial {\phi'}}{\partial {r}} e^{-\mu r} - {\mu \phi' e^{-\mu r}} \right) \right) + 2 \frac{1}{{r}} \left( \frac{\partial {\phi'}}{\partial {r}} e^{-\mu r} -\mu \phi' e^{-\mu r} \right)  - {\mu^2 \phi' e^{-\mu r}} \\ &=\frac{\partial^2 {{\phi'}}}{\partial {{r}}^2} e^{-\mu r} -\mu \frac{\partial {\phi'}}{\partial {r}} e^{-\mu r} -\mu \frac{\partial {\phi'}}{\partial {r}} e^{-\mu r} + 2 \frac{1}{{r}} \left( \frac{\partial {\phi'}}{\partial {r}} e^{-\mu r} -\mu \phi' e^{-\mu r} \right) \\ &=\left( \frac{\partial^2 {{\phi'}}}{\partial {{r}}^2} + 2 \frac{1}{{r}} \frac{\partial {\phi'}}{\partial {r}} \right) e^{-\mu r} -2 \mu \frac{\partial {\phi'}}{\partial {r}} e^{-\mu r} -2 \mu \frac{1}{{r}} \phi' e^{-\mu r} \\ &=(\boldsymbol{\nabla}^2 \phi') e^{-\mu r}- 2 \mu \left( \frac{\partial {\phi'}}{\partial {r}} + \frac{1}{{r}} \phi' \right) e^{-\mu r} \\ &=(\boldsymbol{\nabla}^2 \phi') e^{-\mu r}- 2 \frac{\mu}{r} \left( \frac{\partial {(r \phi')}}{\partial {r}} \right) e^{-\mu r} \\ \end{aligned}

There’s an extra term here that doesn’t show up in (32.105) with this transformation. Can that be argued away somehow?

\item Page 589. (32.107). minor: i \mathbf{k}_f . instead of i \mathbf{k}_f \cdot
\item Page 589. (32.108). minor: i \mathbf{q} . instead of i \mathbf{q} \cdot
\item Page 590. (after 32.118). minor: i \mathbf{q} . instead of i \mathbf{q} \cdot
\item Page 591. (32.124). minor: i \mathbf{q} . instead of i \mathbf{q} \cdot
\end{itemize}

NOTE: up to commit efc6cd3bfee91f43949016f8ba851de273e4fa8d of these notes emailed to Desai May 10, 2011.

Chapter 33.

\begin{itemize}
\item Page 597. (before 33.5). Last paragraph references chapter 12. Chapter 32 meant here? (or chapter 4).
\end{itemize}

Chapter 35.

\begin{itemize}
\item Page 635. (35.46). minor: Bold on gamma.
\item Page 636. (35.50). minor: Bold on gamma.
\item Page 636-638. (35.51-35.58). minor: incomplete notation switch. This chapter uses E_p instead of {\left\lvert{E}\right\rvert}, but many formulas on these pages continue to use the {\left\lvert{E}\right\rvert} notation from chapter 33, even mixing the two in some places.
\item Page 643. (35.107). e_\mu^{\nu} should be e_\mu^{.\nu}. There are also some missing positional indicators in (35.105) and (35.106).
\item Page 643. (35.114). \bar{\psi}'(x')\psi(x') should be \bar{\psi}'(x')\psi'(x').
\item Page 645. (35.134). Wrong sign on \gamma_5. It should be -i \gamma^1 \gamma^2 \gamma^3 \gamma^4.
\end{itemize}

Chapter 36.

\begin{itemize}
\item Page 648. (36.16-17). \hbars should be omitted for consistency.
\item Page 648. (36.16-17). It appears that the - e \boldsymbol{\sigma}' should be + e \boldsymbol{\sigma}'.
\end{itemize}

References

[1] BR Desai. Quantum mechanics with basic field theory. Cambridge University Press, 2009.

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