The Quantum Theory of the Electron
1928 · paper · Physics
Seeks to explain why electrons show doubled numbers of atomic states without assuming a spinning-electron model. Reports that the simplest Hamiltonian for a point electron satisfying both relativity and quantum transformation theory accounts for these phenomena without further assumption.
Checked: PDF from mathweb.ucsd.edu, fetched 4 Oct 2026. Page numbers are those of that PDF.
2 findings
Substance The classical Hamiltonian "F ≡ (W/c + e/c·A0)² + (p + e/c·A)² + m²c²" and equation (1) "[(ih ∂/c∂t + e/c·A0)² + Σr(−ih ∂/∂xr + e/c·Ar)² + m²c²]ψ = 0" are printed with a plus sign on the energy term, but §2 says (1) reduces with no field to "(−p0² + p² + m²c²)ψ = 0" (3), and p. 619 gives "−(p0 + e'A0)² + (p + e'A)² + m²c² + …" as differing from (1) only by the two spin terms; the first squared term of F and of (1) needs a minus sign.
- p. 3 §1, definition of F (display before 'where p is the momentum vector')
+ m2c2,
- p. 3 §1, equation (1)
= 0, (1)
- p. 5 §2, equation (3)
when equation (1) reduces to
- p. 11 §4, expanded form of (15) and the sentence after it
This differs from (1) by the two extra terms
Judge's note: Read F and equation (1) on an enlarged render of PDF page 3: no minus sign precedes the first bracket in either. Read (3) on page 5, with p0 = W/c = ih ∂/c∂t, and the expanded (15) on page 11. Setting A0 = A = 0 in (1) as printed gives (+p0² + p² + m²c²)ψ, not (3). Equation (5) on page 5 also yields −p0², and page 11 says its expression "differs from (1) by the two extra terms", which is true only if the first term of (1) is negative. No imaginary-time convention is introduced in §1 that would absorb the sign. The rest of the paper consistently uses the minus sign, so the fix is evident.
Second check: another model agreed
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Substance The substitution "χ = B^(−½)χ1", said to "eliminate the ∂χ/∂r term" of (26), does not do so for (26) as printed (last term "− (1/Bh)(∂V/∂r)(∂/∂r + (j+1)/r)χ"), and (27) as printed ("+[(1/Bh)(j/r)∂V/∂r − ½(1/Bh)∂²V/∂r² + ¼(1/B²h²)(∂V/∂r)²]χ1") does not follow from (26); (26) and (27) disagree in the sign of the 1/(Bh) terms, on which the sign of the first-order correction (28), (28') rests.
- p. 15 §6, display between (26) and (27)
To do this we must eliminate the
- p. 15 §6, equation (26)
(26)
- p. 15 Equation (26), last term
If one neglects the last term, which is small on account of B being large
- p. 16 First-order correction after (27), and conclusion
The correction is now, to the first order of accuracy
Only the authors can say what is right here.
Judge's note: Read pages 14-16 on the page images and an enlarged render of page 15 (exponent −½, minus on the last term of (25) and (26), bracket of (27) with leading plus and coefficient ¼). Reworked it myself. From the two component equations on page 14 and hB = p0 + V + mc: (∂/∂r + j/r)ψβ = Bψα and ∂ψα/∂r = (1/h)[−(p0+V−mc)ψβ + (jh/r)ψα]; differentiating gives (25) with the minus sign as printed, and ψβ = rχ gives (26) as printed. So (25) and (26) follow from page 14. With b = B'/B = (1/Bh)∂V/∂r and χ = B^a·χ1, the coefficient of ∂χ1/∂r beyond 2/r is (2a − 1)b: zero for a = +½, and −2b for the printed a = −½. With a = +½ the χ1 coefficient is −(j/r)b + ½B''/B − ¾b², the opposite sign to the printed bracket in its first two terms, and ¾ rather than ¼ in the third. The printed (27) is exactly what a = −½ gives if the last term of (26) had a plus sign: first-derivative coefficient 2a + 1 = 0, and χ1 coefficient +(j/r)b − ½B''/B + ¼b². So the printed exponent and (27) are consistent with each other but not with (25)/(26), which in turn follow from page 14. The first-order correction on page 16 and (28), (28') are taken from the printed (27), so their overall sign relative to (26) is reversed. (28) and (28') have opposite signs to each other either way, so the pair still matches the pair of Pauli-Darwin values up to this overall sign. Only the author could say which convention was intended; I did not check the sign of the quoted Pauli-Darwin term against those papers.
Second check: another model agreed
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1 minor (typos and the like; most checked by the judge only)
Typo In the transition-current expression "I_mn = −(e/2m){−ih(ψ_m grad ψ̄_n − ψ̄_n grad ψ_m) + 2(e/c)·A_m ψ_m ψ̄_n}" the vector potential carries a subscript m; it should be A, as in the companion ρ_mn expression, which has "2eA0 ψ_m ψ̄_n".
- p. 3 §1, expression for I_mn (Gordon–Klein current)
are to be interpreted as the charge and current associated with the transition
Judge's note: Enlarged render of PDF page 3: the bold A in the last term of I_mn clearly has a subscript m. The vector potential is the external field A, written without a state index everywhere else on the page (F, equation (1), ρ_mn). m and n label the two solutions here, and the equation is a vector equation, so m is not a component label. Stray subscript, cosmetic.
Second check: none (judge only)