Quantisierung als Eigenwertproblem
1926 · paper · Physics
Shows for the hydrogen atom that the usual quantization rule can be replaced by a variational requirement on a wave function, with no mention of whole numbers. Integer values arise naturally, like nodes of a vibrating string, and the discrete spectrum matches the Balmer terms.
1 finding
Wording Page 10: "weil sie im Unendlichen nur wie 1/r, ∂ψ/∂r auf einer großen Kugel also nur wie 1/r² gegen Null geht. Das Oberflächenintegral (6) bleibt daher gerade noch von der Ordnung des δψ im Unendlichen" — but page 7 says the E > 0 solutions go to zero like 1/r "unter beständigen Oszillationen", and for an oscillating 1/r decay ∂ψ/∂r falls off only like 1/r, so the surface integral would be of order r·δψ.
- p. 10 Paragraph beginning 'Zweitens', statement about ∂ψ/∂r
einer großen Kugel also nur wie 1/r² gegen Null geht
- p. 7 Italic summary for case 1, E > 0
ständigen Oszillationen wie 1/r gegen Null gehen
- p. 10 Sentence following, consequence for surface integral (6)
Ordnung des δψ im Unendlichen
- p. 10 §1, paragraph beginning 'Zweitens ist jetzt daran zu erinnern'
auf einer großen Kugel also nur wie 1/r² gegen Null geht
Only the authors can say what is right here.
Judge's note: Viewed pages 7 and 10; did not open page 11. Page 7 states that c1, c2 are purely imaginary for E > 0, that the exponentials in (17) are bounded periodic functions, and that the solutions go to zero like 1/r under persistent oscillations. Page 10 infers ('also') that ∂ψ/∂r goes like 1/r². Differentiating sin(kr)/r gives k·cos(kr)/r − sin(kr)/r², whose leading term is of order 1/r; the 1/r² rate holds only for a monotone 1/r decay. With a surface element of order r², integral (6) is then of order r·δψ, not of order δψ as printed. I found no statement on these pages restricting the claim to an envelope or an average. The qualitative conclusion, that an extra condition on δψ at infinity is needed, still holds, so I rate this as a misdescription rather than a substantive error. It is a slip in the author's reasoning, not a misprint, so it goes to author query.
Second check: another model agreed
Report a problem with this finding
1 minor (typos and the like; most checked by the judge only)
Typo Equation (1'') prints the Coulomb term as "(E + ²/r) ψ² = 0", with a bare superscript 2 over r; the letter e is missing (should be e²/r, as in equations (3), (4) and (5) on the same page).
- p. 2 Equation (1'')
(1'')
- p. 2 Equation (3), second line, same term printed correctly
das Integral erstreckt über den ganzen Raum
Judge's note: Viewed the page 2 image. In (1'') the numerator over r is only a superscript 2 with no base letter, while (3), (4) and (5) on the same page clearly print e²/r, and the sentence above introduces e as the electron's charge. The fix is evident. I cannot tell from this scan whether the letter failed to print in the original or dropped out only in this scanned copy; either way it is absent from the PDF as supplied.
Second check: none (judge only)