2,900 research outputs found
Kuhn Losses Regained: Van Vleck from Spectra to Susceptibilities
We follow the trajectory of John H. Van Vleck from his 1926 Bulletin for the
National Research Council (NRC) on the old quantum theory to his 1932 book, The
Theory of Electric and Magnetic Susceptibilities. We highlight the continuity
of formalism and technique in the transition from dealing with spectra in the
old quantum theory to dealing with susceptibilities in the new quantum
mechanics. Our main focus is on the checkered history of a numerical factor in
the Langevin-Debye formula for the electric susceptibility of gases. Classical
theory predicts that this factor is equal to 1/3. The old quantum theory
predicted values up to 14 times higher. Van Vleck showed that quantum mechanics
does away with this "wonderful nonsense" (as Van Vleck called it) and restores
the classical value 1/3. The Langevin-Debye formula thus provides an
instructive example of a Kuhn loss in one paradigm shift that was regained in
the next. In accordance with Kuhn's expectation that textbooks sweep Kuhn
losses under the rug, Van Vleck did not mention this particular Kuhn loss
anywhere in his 1926 NRC Bulletin (though he prominently did flag a Kuhn loss
in dispersion theory that had recently been regained). Contrary to Kuhn's
expectations, however, he put the regained Kuhn loss in susceptibility theory
to good pedagogical use in his 1932 book. Kuhn claimed that textbooks must
suppress, truncate, and/or distort the prehistory of their subject matter if
they are to inculcate the exemplars of the new paradigm in their readers. This
claim is not borne out in this case. Because of the continuity of formalism and
technique that we draw attention to that Van Vleck could achieve his
pedagogical objectives in his 1932 book even though he devoted about a third of
it to the treatment of susceptibilities in classical theory and the old quantum
theory in a way that matches the historical record reasonably well.Comment: This paper will be published in: Massimiliano Badino and Jaume
Navarro (eds.), Research and Pedagogy: A History of Early Quantum Physics
through its Textbooks, Berlin: Edition Open Access, forthcoming. This volume
is part of a larger project on the history of quantum physics of the Max
Planck Institute for History of Science in Berli
The trouble with orbits: the Stark effect in the old and the new quantum theory
The old quantum theory and Schr\"odinger's wave mechanics (and other forms of
quantum mechanics) give the same results for the line splittings in the
first-order 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 Schr\"odinger 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
condition
Pascual Jordan's resolution of the conundrum of the wave-particle duality of light
In 1909, Einstein derived a formula for the mean square energy fluctuation in
black-body radiation. This formula is the sum of a wave term and a particle
term. In a key contribution to the 1925 Dreimaennerarbeit with Born and
Heisenberg, Jordan showed that one recovers both terms in a simple model of
quantized waves. So the two terms do not require separate mechanisms but arise
from a single consistent dynamical framework. Several authors have argued that
various infinities invalidate Jordan's conclusions. In this paper, we defend
Jordan's argument against such criticism. In particular, we note that the
fluctuation in a narrow frequency range, which is what Jordan calculated, is
perfectly finite. We also note, however, that Jordan's argument is incomplete.
In modern terms, Jordan calculated the quantum uncertainty in the energy of a
subsystem in an energy eigenstate of the whole system, whereas the thermal
fluctuation is the average of this quantity over an ensemble of such states.
Still, our overall conclusion is that Jordan's argument is basically sound and
that he deserves credit for resolving a major conundrum in the development of
quantum physics.Comment: This paper was written as part of a joint project in the history of
quantum physics of the Max Planck Institut fuer Wissenschaftsgeschichte and
the Fritz Haber Institut in Berli
Efficient dielectric matrix calculations using the Lanczos algorithm for fast many-body implementations
We present a implementation that assesses the two major bottlenecks
of traditional plane-waves implementations, the summations over conduction
states and the inversion of the dielectric matrix, without introducing new
approximations in the formalism. The first bottleneck is circumvented by
converting the summations into Sternheimer equations. Then, the novel avenue of
expressing the dielectric matrix in a Lanczos basis is developed, which reduces
the matrix size by orders of magnitude while being computationally efficient.
We also develop a model dielectric operator that allows us to further reduce
the size of the dielectric matrix without accuracy loss. Furthermore, we
develop a scheme that reduces the numerical cost of the contour deformation
technique to the level of the lightest plasmon pole model. Finally, the use of
the simplified quasi-minimal residual scheme in replacement of the conjugate
gradients algorithm allows a direct evaluation of the corrections at
the desired real frequencies, without need for analytical continuation. The
performance of the resulting implementation is demonstrated by
comparison with a traditional plane-waves implementation, which reveals a
500-fold speedup for the silane molecule. Finally, the accuracy of our
implementation is demonstrated by comparison with other calculations
and experimental results.Comment: 19 pages, 2 figure
Presentism and Relativity
In this critical notice we argue against William Craig's recent attempt to reconcile presentism (roughly, the view that only the present is real) with relativity theory. Craig's defense of his position boils down to endorsing a neo-Lorentzian interpretation of special relativity. We contend that his reconstruction of Lorentz's theory and its historical development is fatally flawed and that his arguments for reviving this theory fail on many counts
Quantization Conditions, 1900–1927
We trace the evolution of quantization conditions from Planck's introduction of a new fundamental constant (h) in his treatment of blackbody radiation in 1900 to Heisenberg's interpretation of the commutation relations of modern quantum mechanics in terms of his uncertainty principle in 1927
Sleeping Beauty on Monty Hall
We present a game show that we claim can serve as a proxy for the notorious Sleeping Beauty Problem. This problem has divided commentators into two camps, 'halfers' and 'thirders'. In our game show, the potential awakenings of Sleeping Beauty, during which she will be asked about the outcome of the coin toss that determined earlier how many times she is awakened and asked, are replaced by potential contestants, deciding whether to choose heads or tails in a bet they will get to place if chosen as contestants on the outcome of the coin toss that determined earlier how many of them are chosen as contestants. This game show bears out the basic intuition of the thirders. Our goal in this paper, however, is not to settle the dispute between halfers and thirders but to draw attention to our game-show proxy itself, which realizes a version of the Sleeping Beauty Problem without the ambiguities plaguing the original. In this spirit, we design similar game-show proxies for variations on the Sleeping Beauty Problem with stochastic experiments other than a coin toss. We do the same for a variation in which Sleeping Beauty must decide upon being awakened whether or not to switch doors in the famous Monty Hall Problem and have the number of awakenings during which she gets to make that decision depend on the door she picked before she was put to sleep
Sleeping Beauty on Monty Hall
Inspired by the Monty Hall Problem and a popular simple solution to it, we present a simple solution to the notorious Sleeping Beauty Problem. We replace the awakenings of Sleeping Beauty by contestants on a game show like Monty Hall’s and we increase the number of awakenings/contestants in the same way that the number of doors in the Monty Hall Problem is increased to make it easier to see what the solution to the problem is. We show that the Sleeping Beauty Problem and variations on it can be solved through simple applications of Bayes’s theorem. This means that we will phrase our analysis in terms of credences or degrees of belief. We will also rephrase our analysis, however, in terms of relative frequencies. Overall, our paper is intended to showcase, in a simple yet non-trivial example, the efficacy of a tried-and- true strategy for addressing problems in philosophy of science, i.e., develop a model for the problem and vary its parameters. Given that the Sleeping Beauty Problem, much more so than the Monty Hall Problem, challenges the intuitions about probabilities of many when they first encounter it, the application of this strategy to this conundrum, we believe, is pedagogically useful
Quantization Conditions, 1900–1927
We trace the evolution of quantization conditions from Planck's introduction of a new fundamental constant (h) in his treatment of blackbody radiation in 1900 to Heisenberg's interpretation of the commutation relations of modern quantum mechanics in terms of his uncertainty principle in 1927
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