On the verge of Umdeutung in Minnesota: Van Vleck and the correspondence principle. Part two

Duncan, Anthony; Janssen, Michel

doi:10.1007/s00407-007-0009-3

Cited by 21 publications

(9 citation statements)

References 107 publications

(278 reference statements)

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“…The one other accomplishment the authors explicitly identify as providing "a strong argument in favour of the theory" is their derivation of the Kramers dispersion formula, "otherwise obtained only on the basis of correspondence considerations" (3M, p. 333). Since the new theory grew directly out of these correspondence considerations (Duncan and Janssen, 2007), it is not terribly surprising that it correctly reproduces this formula. The recovery of the Einstein fluctuation formula, which played no role in the construction of the theory, constitutes much more striking evidence.…”

Section: Jordan's Results As Evidence For Matrix Mechanicsmentioning

confidence: 87%

“…Address: Tate Laboratory of Physics, 116 Church St. SE, Minneapolis, MN 55455, USA, Email: janss011@umn.edu 1 See (Duncan and Janssen, 2007) both for an account of what led Heisenberg to this idea and for further references to the extensive historical literature on this subject. Both our earlier paper and this one are built around the detailed reconstruction and elementary exposition of one key result-Van Vleck's derivation of the Kramers dispersion formula in the former (secs.…”

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confidence: 99%

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Pascual Jordan's resolution of the conundrum of the wave-particle duality of light

Duncan

Janssen

2008

Studies in History and Philosophy of Science Part B: Studies in

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In 1909, Einstein derived a formula for the mean square energy fluctuation in a subvolume of a box filled with black-body radiation. This formula is the sum of a wave term and a particle term. Einstein concluded that a satisfactory theory of light would have to combine aspects of a wave theory and a particle theory. In a key contribution to the 1925 Dreimännerarbeit with Born and Heisenberg, Jordan used Heisenberg's new Umdeutung procedure to quantize the modes of a string and argued that in a small segment of the string, a simple model for a subvolume of a box with black-body radiation, the mean square energy fluctuation is described by a formula of the form given by Einstein. The two terms thus no longer require separate mechanisms, one involving particles and one involving waves. Both terms arise from a single consistent dynamical framework. Jordan's derivation was later criticized by Heisenberg and others and the lingering impression in the subsequent literature is that infinities of one kind or another largely invalidate Jordan's conclusions. In this paper, we carefully reconstruct Jordan's derivation and reexamine some of the objections raised against it. Jordan could certainly have presented his argument more clearly. His notation, for instance, fails to bring out that various sums over modes of the string need to be restricted to a finite frequency range for the final result to be finite. His derivation is also incomplete. In modern terms, Jordan only calculated the quantum uncertainty in the energy of a segment of the string in an eigenstate of the Hamiltonian for the string as a whole, while what is needed is the spread of this quantity in a thermal ensemble of such states. These problems, however, are easily fixed. Our overall conclusion then is that Jordan's argument is basically sound and that he deserves credit for resolving a major conundrum in the development of quantum physics. 1 See (Duncan and Janssen, 2007) both for an account of what led Heisenberg to this idea and for further references to the extensive historical literature on this subject. Both our earlier paper and this one are built around the detailed reconstruction and elementary exposition of one key result-Van Vleck's derivation of the Kramers dispersion formula in the former (secs. 5.1-5.2 and 6.2) and the derivation of the mean square energy fluctuation in the small segment of a string in the Dreimännerarbeit in the latter (sec. 3 below).

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Section: Jordan's Results As Evidence For Matrix Mechanicsmentioning

confidence: 87%

mentioning

confidence: 99%

Pascual Jordan's resolution of the conundrum of the wave-particle duality of light

Duncan

Janssen

2008

Studies in History and Philosophy of Science Part B: Studies in

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“…The improved accessibility of texts and the enlargement of electronic repositories have made it much easier to discover new connections, patterns and influences, and to unearth new stories and new characters. For instance, just when it seemed that we knew all that was worth knowing about Heisenberg's Umdeutung paper, Michel Janssen and Tony Duncan managed to demonstrate that the American physicist John Van Vleck came, very close to the same result (Duncan and Janssen, ,b). Van Vleck was working in Minnesota, a place not quite as glamorous as Göttingen, although he was able to ground his approach on the imposing American tradition of celestial mechanics.…”

Section: Sourcesmentioning

confidence: 99%

What Have the Historians of Quantum Physics Ever Done for Us?

Badino

2016

Centaurus

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“…In the mid-1920s, a number of physicists working on the old quantum theory became increasingly suspicious of the notion of electron orbits. In the transition from the old quantum theory to both matrix mechanics and wave mechanics, orbits were discarded altogether (see Duncan and Janssen (2007) for the case of matrix mechanics). What we have shown in this paper, using the account of the Stark effect in the old and the new quantum theory as a striking example, is that orbits had become highly problematic well before the developments of the mid-1920s.…”

Section: Conclusion: Stark Contrasts Between the Old And The New Quan...mentioning

confidence: 99%

The trouble with orbits: The Stark effect in the old and the new quantum theory

Duncan¹,

Janssen²

2014

Studies in History and Philosophy of Science Part B: Studies in

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The old quantum theory and Schrödinger's wave mechanics (and other forms of quantum mechanics) give the same results for the line splittings in the firstorder Stark effect in hydrogen, the leading terms in the splitting of the spectral lines emitted by a hydrogen atom in an external electric field. We examine the account of the effect in the old quantum theory, which was hailed as a major success of that theory, from the point of view of wave mechanics. First, we show how the new quantum mechanics solves a fundamental problem one runs into in the old quantum theory with the Stark effect. It turns out that, even without an external field, it depends on the coordinates in which the quantum conditions are imposed which electron orbits are allowed in a hydrogen atom. The allowed energy levels and hence the line splittings are independent of the coordinates used but the size and eccentricity of the orbits are not. In the new quantum theory, this worrisome non-uniqueness of orbits turns into the perfectly innocuous non-uniqueness of bases in Hilbert space. Second, we review how the so-called WKB (Wentzel-Kramers-Brillouin) approximation method for solving the Schrödinger equation reproduces the quantum conditions of the old quantum theory amended by some additional half-integer terms. These extra terms remove the need for some arbitrary extra restrictions on the allowed orbits that the old quantum theory required over and above the basic quantum conditions. 1

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On the verge of Umdeutung in Minnesota: Van Vleck and the correspondence principle. Part two

Cited by 21 publications

References 107 publications

Pascual Jordan's resolution of the conundrum of the wave-particle duality of light

Pascual Jordan's resolution of the conundrum of the wave-particle duality of light

What Have the Historians of Quantum Physics Ever Done for Us?

The trouble with orbits: The Stark effect in the old and the new quantum theory

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