Über quantentheoretische Umdeutung kinematischer und mechanischer Beziehungen

• p. 3 First displayed equation

  ν (n, n – α) = {W (n) – W (n – α)}

• p. 8 §2, energy-integral paragraph

  ∆W = h ⋅ν

Judge's note: The p. 3 image shows nothing in front of the braces. The classical counterpart directly below has α·(1/h)·dW/dn, and p. 8 states "ΔW = h·ν". The paper does not set h = 1 anywhere, so the displayed formula is dimensionally inconsistent with the rest of the paper. Evident fix: prefactor 1/h.

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• p. 9 First displayed formula for M

  | a(n, n+α) |² ν (n, n−α) / (ν² (n, n−α) − ν²)

• p. 9 Second displayed formula for M

  | a(n, n−α) |² ν (n, n−α)

• p. 9 first displayed formula for M

  | a(n, n+α) |² ν(n, n−α)

Judge's note: Viewed the crop and the full page. Both numerators in the first formula carry |a(n,n+α)|², while the second term's frequency and denominator use (n,n−α). The limiting formula on the next line and eq. (16) on p. 8 both pair |a(n,n−α)|² with ν(n,n−α), so the stated limit does not follow from the first formula as printed. Evident fix: a(n,n−α) in the second term.

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• p. 9 Quantum-theoretical Ansatz after eq. (17), second line

  …, λτ−1 a (n, n − 1) cos ω (n, n − 1) t, …

• p. 9 quantum Ansatz, last displayed line

  λτ−1 a (n, n − 1) cos ω (n, n − 1) t

Judge's note: Page 9 image shows the general term with (n, n−1) in both a and ω. The τ=1 term a(n,n−1) cos ω(n,n−1)t is already listed without a λ factor, the classical series has λ^(τ−1) a_τ cos τωt, and eq. (21) on p. 11 uses a(n, n−τ). The same classical line also prints "λ a0 cos 2ωt" where a2 is expected.

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• p. 10 Eq. (19), first line

  ω0² a0(n) + [a²(n+1, n) + a²(n, n−1)] / 2 = 0

• p. 10 Eq. (18), first line

  ω0² a0(n) + a1²(n) / 2 = 0

• p. 10 Eq. (19), first line

  a²(n+1, n) + a²(n, n−1) / 2

Only the authors can say what is right here.

Judge's note: Page 10 image shows denominator 2. Reworked: with x containing a(n,n−1) cos ω(n,n−1)t, the exponential amplitudes are a/2, so by (7) the constant term of x² is a²(n,n−1)/4 + a²(n+1,n)/4. For large n both a² tend to a1², giving a1²/2 as in (18); as printed the limit is a1², twice the classical value. Lines 3 and 4 of (19) do reduce to (18) (a·a/2 → a1²/2; two terms each a1a2/2 → a1a2), so the normalisation is not the cause.

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• p. 11 Classical Ansatz after ẍ + ω0²x + λx³ = 0

  x = a1 cos ω t + λ a3 cos 2ω t + λ3 a5 cos 5ω t + …

• p. 12 Quantum Ansatz

  λ a (n, n – 3) cos ω (n, n – 3) t

• p. 11 cubic anharmonic Ansatz

  λ a3 cos 2ω t + λ3 a5 cos 5ω t

• p. 11 Classical Ansatz for x''+ω0²x+λx³=0

  λ a3 cos 2ω t

Judge's note: Page 11 image shows 'λ a3 cos 2ωt'. The series runs over odd harmonics (a1, a3, a5 with cos ωt, cos 5ωt), and the quantum Ansatz on p. 12, stated to be analogous, has λ a(n,n−3) cos ω(n,n−3)t. The λ³ on the a5 term is judged separately in C020.

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• p. 11 Classical Ansatz after ẍ + ω0²x + λx³ = 0

  λ3 a5 cos 5ω t

Judge's note: Page 11 image shows a superscript 3 on λ. For a λx³ perturbation each further odd harmonic enters one order of λ higher (a1 at λ⁰, a3 at λ¹, a5 at λ²). The earlier Ansatz on p. 9 uses λ^(τ−1) per harmonic step, and the a are said to begin with λ-free terms. Kept separate from C019, which covers the cos 2ωt slip in the same line.

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• p. 12 Eq. (24)

  ω (n, n – 1) = ω0 + λ ⋅ 2nh / (8π ω0² m)

• p. 12 Eq. (27), λ term

  3(n² + n + ½)h² / (8 ⋅ 4π² ω0² ⋅ m)

• p. 12 Eq. (24), λ term

  2nh

• p. 12 Eq. (27), λ term

  3(n² + n + ½) h²

Judge's note: Page 12 image shows numerator 2nh in (24). Reworked: the λ term of (27) is 3(n²+n+½)h²/(32π²ω0²m); W(n)−W(n−1) = 3·2n·h²/(32π²ω0²m); times 2π/h gives 3nh/(8πω0²m). The λ² term checks: 17(3n²−3n+1) + (51/2)(2n−1) + 59/2 = 51n²+21 = 3(17n²+7), and 2π/(512π³) = 1/(256π²), which matches (24). So only the first-order numerator disagrees, and the text explicitly ties (27) to (24). Eq. (25) also carries 3nh.

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• p. 12 Eq. (25)

  − λ2 ⋅ 3h² / (256 ω0⁵ m² π²) ,

• p. 12 Eq. (25), trailing term

  − λ² ⋅ 3h² / (256 ω0⁵ m² π²)

• p. 12 sentence before Eq. (24)

  up to the approximation λ2 (λ, resp.)

Only the authors can say what is right here.

Judge's note: Page 12 image shows the term after the closing bracket of (25). Dimensions reworked: from ẍ ~ λx³, [λ] = L⁻²T⁻²; [h/m] = L²T⁻¹; so [λ²h²/(ω0⁵m²)] = L⁻⁴T⁻⁴·L⁴T⁻²·T⁵ = T⁻¹, while a(n,n−1) is a length. The in-bracket term λ·3nh/(16πω0³m) is dimensionless, as it should be. The trailing term has no n-dependence, the bracket already ends in '+ ⋯', and the text says a is taken only to order λ ('λ² (λ, resp.)'). It appears to be a duplicated fragment of (24); the authors or translator should confirm it is to be deleted.

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• p. 14 Eq. (33), second line

  ½ a0² = b1 b−1

• p. 14 Eq. (31)

  ½ a (n, n – 1; m, m) a (n − 1, n – 2; m, m)

Only the authors can say what is right here.

Judge's note: Page 14 image shows ½ on the second line of (33). Reworked: the classical limit of (31) is ½a0·a0 = b−1b1 + b1b−1 = 2b1b−1, i.e. a0²/4 = b1b−1. Check against the p. 15 solutions at large n: a0² ≈ a²(n+m)(n−m)/n², b1² ≈ a²(n+m)²/(4n²), b−1² ≈ a²(n−m)²/(4n²), so b1b−1 = a²(n²−m²)/(4n²) = a0²/4. Concrete case n=10, m=6 with a=1 in the large-n forms: a0² = 0.64, b1b−1 = 16·4/400 = 0.16 = a0²/4, not a0²/2. The first line of (33) does hold with the same values (0.32 + 0.64 + 0.04 = 1), so a different normalisation of b does not explain it.

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• p. 14 Eq. (33), third line

  2π m (b²+1 − b²−1) = (m + const)h

• p. 14 Eq. (32)

  2π m {b2 (n, n – 1; m, m – 1) ω (n, n – 1) − b2 (n, n – 1; m – 1, m) ω (n, n – 1)}

Only the authors can say what is right here.

Judge's note: Page 14 image shows (32) with ω(n,n−1) multiplying both b² terms and the third line of (33) without any ω. Dimensions: [2π·m·b²] = M·L², while [h] = M·L²·T⁻¹, so a factor of frequency is needed for the classical relation to correspond to (32). The same line writes b₊₁ where the other lines of (33) and the following sentence write b1 (cosmetic).

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• p. 14 Eq. (30), second line

  b2 (n – 1, n; m – 1, m)

Judge's note: Viewed the crop and full page 14. The first line of (30) carries the three (n, n−1) terms and the second line the three upward-neighbour terms; two of those are printed with (n+1, n) and the middle one with (n−1, n). Every other amplitude on pages 14-15 is written with the larger n first. Read as the reversed form of b(n, n−1; ...), the term would duplicate a first-line contribution and leave the n+1 set incomplete. I judge it an index misprint with an evident fix.

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• p. 2 §1, parenthetical legend after the radiation formulas

  t means the distance from the electron to the reference point

Judge's note: Viewed the crop and the full page. The legend glyph is a Fraktur t, visibly different from the Fraktur r in [𝔯[𝔯 v̇]] and the italic r in the denominators. No t appears in the formulas, and r is left undefined. Evident fix: 𝔯 (or r).

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• p. 3 Paragraph before eq. (1)

  the two variables n and a

• p. 9 Between the first and second M formulas

  and thus for ν ≫ ν (n, n + a):

• p. 9 condition before second M formula

  ν (n, n + a)

• p. 7 Eq. (13), exponent

  i a ωn t

Judge's note: On pp. 3 and 9 the glyph is an italic Latin a, clearly different from the α in the adjacent formulas (crop C005-b). In eq. (13) on p. 7 the exponent is small on the image, but it reads a rather than α, and the text layer gives "i a"; the derivative line has the prefactor iαω_n, so α is meant. The letter a is also the amplitude symbol on pp. 7–9, so the slip is confusing.

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• p. 4 Eq. (3)

  Bα (n) ei ω (n) βt =

• p. 4 Eq. (5)

  Bβ (n) e iω(n)βt

• p. 4 Eq. (3), left-hand side

  Bα (n) ei ω (n) βt

• p. 4 Eq. (5)

  Bβ (n)

Judge's note: On p. 4, eq. (3) has 𝔅_α(n) on the left while α runs from −∞ to +∞ on the right and the exponent on the left is ω(n)βt; eq. (5) writes 𝔅_β(n). On p. 5, eq. (7) has 𝔅_α(n,n−β) on the left with α summed on the right. A summed index cannot label the result. Fix: 𝔅_β(n) in (3); drop the subscript (or use β) in (7).

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• p. 9 Ansatz after eq. (17)

  x = λ a0 + a1 cos ω t + λ a0 cos 2ω t + λ2 a3 cos 3ω t

• p. 10 Eq. (18), third line

  (−4ω² + ω0²) a2 (n)

• p. 9 §3 Ansatz

  λ a0 cos 2ω t

Judge's note: The p. 9 image shows a0 both as the constant term and as the cos 2ωt coefficient. Eq. (18) on p. 10 uses a2(n) in the (−4ω²+ω0²) line, and the sequence a1, a3, …, a_τ implies a2. The general quantum term repeats the indices (n, n−1) although it carries λ^{τ−1}; the pattern requires (n, n−τ). Both are subscript/index slips with evident fixes.

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