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List of (Suspected) Errata and Formatting Issues

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  • description of the location of the issue: description of the issue

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- [ ] on p. 123 in (45): missing `i` in `\exp`

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  • on p. 123 in (45): missing i in \exp

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List of contributors

  • Peter
  • Margarit
  • Jens

List of (Suspected) Errata, Formatting Issues & Suggestions for the PMI script (2020/21)

General Remarks

  • It would be nice, if all equations were numbered, making referring to them easier.
  • When referencing back to definitions from previous chapers it would be helpful to add a \eqref to the corresponding equation (e.g. when \hat{\bar{H}} is referenced in chapter 8.2`). This makes it also easier to understand when a symbol is to be understood in its most general or approximated version. (At the beginning of chapter 8.2.2 this is done nicely for the CI matrix.)

By Section

1 (downloaded 2.9.2020)

2 (downloaded 2.9.2020)

  • after (6): (??) looks like a corrupted \eqref

3 (downloaded 11.9.2020)

  • (46): missing \perp symbol in delta-dyadic
  • (58): formatting of velocity off (wrong order of _ and \vec suspected)
  • p.32 last equation last line: missing 2 before \vec{A}^2

4 (downloaded 15.9.2020)

  • mode expansion, before (79) & after (81) (pp 37+38) in normalization factor (occurs multiple times over the script): should be V not \sqrt
  • after (80): 2x missing ), missing index in momenta in commutator, \vec{\Pi} used instead of \hat{\Pi}
  • (81): \vec{\Pi} used instead of \hat{\Pi}
  • (81): missing \perp on delta dyadic, the vanishing grad of delta^\perp was already shown in sec. 1

5 (downloaded 17.9.2020)

  • after 5.1 heading: 2x missing )
  • (87): order of operators wrong on the right side of the second equation
  • after (87) in matrix element of exp(i \phi) and also later: first Kronecker delta should be \delta_{n,0} (not with an m) and even better \delta{m, n-1} for the entire matrix element (much simpler and easier to the the hermitian property of cos/sin(\phi) later on)
  • matrix element of sin(\phi) (occurs multiple times over the script): missing factor 1/i

5.2 (downloaded 17.9.2020)

  • in [n, exp(i\phi)]: one \hat{a} too many in the middle & missing minus (-) sign in the last expression
  • (89) and also before that: missing n in \ket
  • (89): missing i in 2nd exponent
  • (90) 2nd line: there should not be a + \hat{a}^\dagger

5.3 (downloaded 19.9.2020)

  • after (98) in <\alpha|\beta>: |\beta|^2 should be in the numerator (not denominator)
  • after (99) 2nd line in completeness relation proof: wrong order of \bra <-> \ket

5.4 (downloaded 19.9.2020)

  • (100): remove the 1/2 in front of t factor in exponential inside the sum
  • after (100) 2nd line: remove one )
  • after (100) 3rd line: \sum should be over m
  • after (101) on page 55 2nd line: missing i in expansion of \cos
  • after (101) on page 55 5th line: missing i in exp (which becomes \cos)
  • after (101) on page 55 5th line: missing \infinity above sum
  • (102): redundant \eq (=) sign

6 (downloaded 19.9.2020)

  • (105): missing \alpha in front of sum

7 (downloaded 22.9.2020)

  • way after (106) on p. 66 after definition of order of 1st coherence function, the trivial clause: there should not be primed coordinates i.e.g^{(1)}(x,t,x,t).
  • after definition of second-order field correlation function: should be G^{(2)}(x_1,t_1,x_2,t_2,x_1,t_1,x_2,t_2) \geq 0 (and not ...x_3,t_3,x_4,t_4)
  • p. 68 example (a) 2nd line: |n should be inside ket
  • p. 68 example (b) 2nd line: square should be in the denominator (not numerator)

8/8.1.1 (downloaded 25.9.2020)

  • first equation inside \sum\limits_n: index of \vec{q} should be \mu and not n
  • (107): missing time derivative
  • (108) on the left: missing i
  • (108) in the first denominator: missing | on the right
  • (108): change dagger to stars
  • (109): missing Laplace operator
  • after (109) on p.71 2nd equation left side right term in sum: missing \delta in numerator and denominator
  • after (109) on p.71 last equation first line left side: derivation should be with respect to \conjugate\{psi}_\sigma (t) (not its time derivative)
  • after (109) on p.71 last equation first line right side: missing integration variable should be x
  • after (109) p.71 last equation line 3 in front of double intergal: missing prefactor \frac{1}{2}
  • after (109) p.71 last equation multiple lines: change dagger to star
  • after (109) p.72 2nd equation last line: missing \prime and ( for second \conjugate{\psi}
  • (this one is optional) last line in text on p.72: missing space after closing )
  • p.73 first set of equations 3rd line right side of anti-commutator: change \Psi into \Pi
  • p.73 before (110) in the text: \dagger not in exponent
  • (110) 2nd line: missing (x) for first \psi
  • p.73 after (110) and anti-commutator relations: first sentence is not finished (may be ... hold for any particle number).
  • p.73 last line in paragraph where One-Body operators is written: change nof to is of (?)
  • p.73 second to last equation right side: shouldn't the \hat be on \vec{x} rather than the t?
  • p.74 second paragraph: the the
  • p.74 2nd equation first line: missing \dagger
  • p.74 4th equation right side: missing \hats

8.1.2 (downloaded 25.9.2020)

  • p.75 last equation:\phi_pshould be \psi_p
  • (suggestion) p.75 last equation: small \omega can be confusing as a abbreviation for the angle coordinates since it looks like it is related to an oscillation. A capitalized \Omega might be less ambiguous here - or just use (\theta, \phi).
  • p.76 2nd equation: remove (r)
  • p.76 2nd paragraph: typo meaningfuel
  • p.76 paragraph before the 3rd equation: change \psi_p to \phi_p
  • p.76 3d equation: q should be in index of Kronecker delta
  • p.76 2nd to last set of equations both lines on the left: remove prime
  • p.76 last equation missing parameter (\vec{x}) for orbit
  • (suggestion) p.76 last equation: It would be nice if an explanation is given why why the orbit function is written only here in front of an annihilation operator. Maybe the commutating properties of \phi_p(x) and \hat{\Psi}(x) could be discussed here?
  • (suggestion) p.76 end of page: It would be nice if the identity \sum\limits_p \conjugate{\phi_p}(\vec{x}^\prime) \phi_p(\vec{x}) = \delta(\vec{x}^\prime - \vec{x}) resulting from the orthonorality could be included here. Without this identity is seems impossible to show that the last two blocks of equations are consistent.
  • p.77 1st equation 3rd row: missing (and ) around \vec{x}^\prime
  • p.77 3rd equation middle term: missing \hat on t
  • p.77 5rd equation 1st row: \vec{\nabla} should be squared
  • p.77 4rd equation: alignment for 2nd row is off
  • p.77 4rd equation last line: missing integration variable \vec{x}
  • p.77 4rd equation last line: change summation index o to p
  • p.77 5th equation 2nd row: missing index s in anihilation operator
  • p.78 1st equation 1st line: missing \laplace operator
  • p.78 1st equation 1st line: missing \ket{\phi_q}
  • p.78 1st equation: The \phi_p are not eigenvectors of this hamiltionian, hence there should be either \hat{\bar{V}} in the hamiltonian or the diagonal elements do not vanish. To see this, go back to p. 75 1st equation.

8.1.3 (downloaded 25.9.2020)

  • p.79 listed item 2. middle of the line: missing \Rightarrow
  • p.79 item 3. 3rd line: missing \hat{c}^\dagger_p on the right side
  • p.79 item 5. 1st sentence: typo of of
  • p.80 item 8. 2nd line: missing \hat{c}_p
  • p.80 item 8. 3rd line lower branch of curly braces: inconsistency in space placement, put space after ,

8.2 (dowloaded 2.12.20)

  • p.82 first equation ket on the right: sould be a \ket{ \{n_p\} } (not \psi)
  • p.82 last paragraph: typo for a one-body operators
  • (suggestion) p.83 1st paragraph: In general, use the notation \vec{n} and \vec{n^\prime} instead of \{n_p\} and \{n_q\} to denote occupation numbers. This will avoid confusion in expressions like n_r = 1. Alternatively, more specific and use expressions like n_{p,r} = 1 instead.
  • (suggestion) p.83 1st paragraph last sentence: mention indices of orbitals s and r explicitly.
  • (suggestion) p.84 last equation 1st line: maybe explain the meaning of \epsilon_i here (it was not obvious that it is from \hat{\bar{H}} which includes the + \hat{\bar{V}} and thus the + \sum_i\bar{V}_{ii} arises later).

8.3 (downloaded 2.12.2020)

  • p.86 1st equation 3rd line: \hat{V_{e-e}} instead of \hat{V_{l-l}}

  • p.88 1st paragraph: typo ... entire set of quantum nunbers \epsilon should be \xi

  • p.88 1st paragraph: typo ... angular momentum nunbers l, m should be l, m_l

  • (suggestion) p.88: recap also the definition of the orbital functions specifically, how the radial functions are related to the entire orbital function incl. the normalizations? The missing factor 4 \pi in integral before (115) supprised me. How is the entire orbital function construced. How is \Omega defined. All this should be clear but notations change and it helps to identifiy them.

  • p.88 send equation: missing complex conjugation of one of the spherical harmonics Y

  • (suggestion) p.88 (115): Explain the meaning of l_2 here.

  • p.88 second to last equation: missing \sum over l_2 and m_2?

  • p.89 first equation: where do the \frac{1}{r} come from? typo?

  • p.89 last equation: L_s should be l_s

  • p.89 after (115) 3r equation: 2x missing )

  • p.89 after (115) 3rd eqaution: convention of Clebsch-Gordon coefficient notation broken (either insert remove the , always) -> remove 4x , in between ms

  • (suggestion): write Clebsch-Gordon coefficients always with , even when it's not conventional to prevent mixed notation

  • p.89 last equation 2nd line: replace 4x \epsilon with \xi

  • p.90 last paragraph: typo for i should be for i \neq j

  • p.91 just before the huge equation: missing , in variational derivative between the \phis

  • p.91 the huge equation 2nd line: missing double integral

  • p.91 the huge equation 2nd + 3rd line: missing 2x (\vec{x}_1)

  • p.91 the huge equation 2nd + 3rd line: missing 2x index not in subscript \vec{x}_2 (should not be a square)

  • p.91 the huge equation 5th line: missing )

  • p.91 the huge equation 5th line in the subscript of the \phi right after the d^3x^\prime: missing , between jand \sigma

  • p.91 the huge equation 7th line: use \Leftrightarrow for equivalences for consistency with implication written as \Rightarrow some lines above

  • p.91 the huge equation last line after the last \phi: missaligned )

  • p.92 (116): missing arguments for spatial 1-RDM function

  • p.92 (116): change angle bracket to ) after second \psi

  • p.94 first equation 1st line after d^3x: remove \star

  • p.94 second equation: operator \hat F not f (since it is denoted like that after this page)

  • p.94 second equation: operator \hat J not j (hats for consistency)

  • p.94 second equation: operator \hat K not K (hats for consistency)

  • p.94 after second equation: operator \hat J not J

  • p.94 after third equation: operator \hat K not k

  • p.94 fourth equation: operator \hat K not k

  • p.94 5th equation: 2x operator \hat J not J

  • p.94 6th equation: 2x operator \hat K not k

  • p.94 before last equation: operator \hat F not f

  • p.94 last equation: operator \hat F not f

  • p.99 last eqautin 2nd to last line: missing )

8.7

  • p.106 top: typo Hatree-Salter
  • p.106 end of this section: unfinished sentence

8.8

  • p.106 last equation: \hat{f}should be \hat{F} (inconsistency see introduction of Fock operator)
  • p.107 2nd paragraph: 2x k_f should be k_F (inconsistency see e.g. pp. 108)
  • p.107 2nd paragraph: \epsilon_f should be \epsilon_F (inconsistency)
  • p.107 2nd equation: 2x k_f should be k_F (inconsistency)
  • p.107 2nd paragraph after \vec{k}^{\prime\prime}: \sigma^{\prime\prime} not \sigma
  • p.107 2nd equation 2nd line: remove the 2
  • p.107 4th equation 1st line: missing i in exponential
  • p.108 2nd equation: missing \vec{k}
  • p.109 top, substituion: x=k_f/k should be x=k^{\prime\prime}/k
  • p.109 1st equation 5th line 2nd term: should be\ln(x-1) (not \ln(1-x))
  • p.109 2nd equation end of 2nd line: missing term +\frac{1}{2} \frac{k_F}{k} (\frac{k_f}{k}+1)
  • p.109 last equation 1st line: missing +1 after (\frac{k_f}{k}) +1
  • p.110 1st equation 7th line: missing 1 beore \ln(1-x)
  • p.110 1st equation 7th line: ( should be big
  • p.110 1st equation 7th line: missing )
  • p.110 1st equation end of 5th line: missing term related to one of the integrals (from \int x \ln(1-x))

9.1

  • p.114 last equation 2nd line: missing \dagger
  • p.115 last equation last line: missing prefactor of 2 (?)
  • p.115 last equation last line: missing i in \exp

9.2

  • (suggestion) p. 118 (129: It would be nice, if the notation style is consistent with the previous chapters, i.e. if the sub-- and superscripts N_{el} and 0 were place inside the ket.
  • p.119 after (130): missing restriction in higher order integral t^prime \leq t
  • (suggestion) p.119 end: Mention that up to 1st order this result for the transition rate is also known as Fermi's golden rule. This makes it easier to find secondary literature.
  • p.119 2nd equation 2nd line: m should be M
  • (suggestion) p.119 2nd equation: The justification for dropped terms with \epsilon is not obvious and is therefore confusing - it also violates the strict meaning of the equals sign. It would be good to give an explanation here.

9.3

  • p.120 1st equation:l not in index, should be N__{el}
  • p.120 2nd equation 1st line: remove - in front of N_{el}}
  • (suggestion)p.121 dipole approx: Maybe be more explicit about the approximation, i.e. mention that \abs{\vec{k}_{in} \vec{x}} \ll 1 and therefore the wavelength of the photon must be large.
  • p.120 dipole approx.: The connection from hat{\vec{p}}} to \hat{\vec{r}} as the perturbation operator is missing. See e.g. here. Note: This is a simple yet important derivation and it was crucial when we found a mistake in the numerical exercises.
  • p.120: typo singl-mode
  • p.120 2nd equation 1st line: remove minus in front of N_{el} in \bra{\Psi_F^{-N_{el}}}
  • p.121 2nd paragraph: typo ev instead of eV

9.4

  • p.122 1st equation: missing index in for \vec{k} in the \exp(i k x)-term
  • p.122 1st paragraph: typo on-photon
  • p.123 before the exercise: typo integrating of better: over
  • p.126 1st equation 2nd to last line: missing \tilde on q in E_q^{3/2} in numerator
  • (suggestion) p.125-126: The notation for the free electron momentum q is inconsistent and confusing: The angular wheight d \Omega_{\vec{q}} for example has no \tilde but the final momentum \tilde{\vec{q}} does. It seems simpler to drop the index in d \Omega and give the momentum q while it is in the kernel pf the integral a \prime or something similar.

10.1

  • p.136 1st equation: \gamma should be \theta
  • p.137 2nd equation: \exp (i \vec{x}\vec{x}) should be \exp (i \vec{k} \vec{x}
  • p.137 1st, 5th, 6th eauqtion: inconsistency of the symbol for the principal quantum number, previously \xi now \epsilon, (this migth appear multiple times thoughout the script)

11.1

  • p.139 2nd equation: nontaion of final state in incinsistent F and f
  • p.148 in paragraph after last equation: N should be n in \bra{\Psi_N^{N_{el}-1}}
  • p.151 caption of figure 16 and 17: typo Satnra
  • p.151 aption of figure 17: typo Koopman'stheorem

12

  • p.152 after 1st equation: \Lambda_{in} should be \lambda_{in}
  • p.152 last equation last line: missing \sum \limits_{p,q}
  • p.153 3rd equation: missing i in \exp
  • p.156 last block of eqautions 1st line: missing ^3 shortly after \Rightarrow`
  • p.156 1st equation last line: 1_s sould b3e 1s`

13

  • p.165 last equation: ,L_{2,3} should be in index
  • p.156 after figure 19: typo that mediated

14

  • p.169 first paragraph: equation reference still incomplete

15

  • p.177 1st equation line 3 first bra: missing -1
  • p.178 1st equation last line in denominator: i \epsilon should not be in superscript
  • p.177 1st equation last line in first ket: turn index p into q
  • p.177 1st equation last line in last ket: turn index q into q^\prime

15.1

  • p.180 1st equation last polarization vector: missplaced conjugation symbol

15.2

  • p.186: typo a t
  • p.186 2nd equation 2nd line: missing summation index
  • p.188 last equation 4th line: sign in front of \omega_{in} should be negative
  • p.188 last equation last line: missing ^3 in big O
  • p.190 typo: differenctial

16

  • p.193 eq. 156: inconsistency: photon state should be on the left
  • p.193 eq. 156: typo remove ; and ,
  • p.195 middle of page: typo \Omega_F should be \omega_F
  • p.196 2nd equation 2nd line: rightmost fraction should be polarization sum
  • p.196 3rd equation 1st line: = should be \approx
  • p.196 3rd equation 2nd line rightmost term: misplaced \tilde on \omega_F
  • p.196 4th equation: missing \sum
  • p.197 typo: missing \ in front of sigma_{0,F}
Edited by Lucas Kemmler
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