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Topic: Poincare's work in quantum mechanics
Replies: 1   Last Post: Aug 18, 2003 10:32 AM

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Mark Yasuda

Posts: 3
Registered: 12/13/04
Poincare's work in quantum mechanics
Posted: Aug 17, 2003 6:22 PM
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Although it's rather well known that Henri Poincare anticipated a number of
results in special relativity prior to Einstein's 1905 publication, it seems
that fewer people are aware that Poincare also played a role in another
revolution in physics at the beginning of the 20th century -- namely, quantum
mechanics (I guess there are some people who might also argue that he
participated in another revolution for his pioneering work in dynamical
systems). Recently, I had the good fortune to come across Russell
McCormmach's excellent article "Henri Poincare and the Quantum Theory" (Isis;
Volume 58 (191); 1967; pages 37-55). Besides discussing Poincare's work, it
offers some fascinating glimpses into the first Solvay Conference that took
place in October and November of 1911. Below are a few (six) excerpts that
I've selected from McCormmach's article, which I highly recommend for people
interested in the historical development of physics during this time period:

1. At the time, Maurice de Broglie remarked to F. A. Lindemann that of all
those present Poincare and Einstein were in a class by themselves (p 40).

2. Lorentz recalled that in the discussions Poincare had shown "all the
vivacity and penetration of his spirit, and that one had admired the facility
with which he entered vigorously into even those questions of physics which
were new to him" (p. 40).

3. ... it was Planck, however, who stimulated Poincare's most penetrating,
questioning spirit .... He twice pressed Planck to give good grounds for
deciding among the several possible ways of decomposing phase space into the
finite elementary areas for the probability calculations. He wanted to know
how the energy of a system of several degrees of freedom might be quantized,
since the one-dimensional quantization procedure was incompatible with
transformations of axes in higher-dimensional systems. Poincare regretted
that as yet there had been no discussion of mechanisms for the interaction of
fixed resonators; for in the absence of any definite mechanism there could be
no exchange of energy between radiations of different frequencies, and
therefore no final equilibrium. Planck had stressed quanta of action rather
than quanta of energy . . . but he did not know what it means to speak of the
conservation of action. Finally, he was skeptical of Planck's new formulation
of the radiation theory, according to which the absorption of energy by the
resonators varies continuously with time (p. 41).

4. In a descriptive essay he spelled out the essence of Planck's theory as it
appeared to him: "A physical system is capable of only a finite number of
distinct states; it jumps from one of those states to another without going
through a continuous series of intermediate states." The image of a physical
system jumping from one discrete state to another put him in a speculative
frame of mind. He considered the possibility that a particle might trace only
certain allowed paths in phase space, shifting discontinuously between them.
And he supposed that the universe as well as an electron ought to experience
quantum jumps. Since there would be no distinguishable instants within the
motionless states between universal jumps, there should exist an "atom of
time." Such were the kinds of ideas going through Poincare's mind shortly
before he died; there was nothing timid or grudging about his late
acquaintance with the quantum theory (p 50).

5. ... above all it was the unquestioned authority of Poincare in
mathematical matters which secured him an attentive audience. Jeans
undoubtedly voiced a majority sentiment when he said that "we shall probably
feel inclined to trust to the accuracy of Poincare's mathematics." (pp 51-52).

6. Whereas Jeans had strongly opposed the quantum theory in Brussels . . ., he
came out vigorously in support of quanta at the Birmingham meeting of the
British Association in September 1913, fourteen months after Poincare's death.
There is no doubt about what caused him to change his mind. Jeans had read
Poincare's paper and been converted by it. ... The French scientist's
arguments had been so completely persuasive that from this time on every
theory would have to "logically involve either the belief that Poincare is
wrong, or the belief that he is right, together with all that this involves.
. . . And Jeans himself felt compelled to accept the quantum hypothesis in its
entirety." (p. 53).

As an undergraduate, I had become aware that Poincare's paper "Sur la
theorie des quanta" (one of the last he ever wrote -- he died prematurely in
1912, while undergoing an operation) had been influential in gaining wider
acceptance for Planck's then-controversial quantum theory of blackbody
radiation. The brief historical excerpt I had read at the time implied that
Poincare proved (essentially) that Planck's theory required the existence of
discrete energy quanta.

After reading McCormmach's article, however, I'm less certain that this is an
accurate statement. Instead, it would appear that Poincare's proof (modulo
some necessary refinements [see page 52 of McCormmach]) was based on some
debatable assumptions. My main questions regard the legitimacy of Poincare's
proposed mechanisms whereby pairs of "resonators" exchanged energy (page 45 of
McCormmach). There are other questions that one can raise as well (as some of
Poincare's contemporaries did). I haven't read Poincare's original paper (my
French not being particularly strong), so I'm wondering if anyone who is
familiar with Poincare's work in quantum mechanics can comment on whether
Poincare's legacy in quantum mechanics is either "both A and B" or just "B"
for the following statements:

(A) A legitimate proof that Planck's theory required the existence of discrete
energy quanta (in spite of working off of a physical foundation that predated
Heisenberg and Schroedinger's work in QM by 13-14 years).

(B) Helped to gain further acceptance for the theory of quantum mechanics
among physicists circa 1912.

- Mark

p.s. - While on the topic of Poincare, can anyone comment on whether there is
general concurrence on whether Perelman's third paper

successfully finishes off the Geometrization conjecture (and thereby the
Poincare conjecture)?

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