# The role of anharmonic oscillator(s) in Heisenberg's 1925 paper

I am talking about the most famous paper of Heisenberg, which I know from the translation of van der Waerden (Sources in Quantum mechanics, North Holland, 1967). After introducing matrix mechanics Heisenberg writes (p. 12):

one could regard our equations as satisfactory...if it were possible to show that this solution agrees with quantum mechanics relationships which are known at present

His main examples are the anharmonic oscillators. On pp. 272-273 he considers a quartic oscillator whose classical equation is $$x''+\omega_0^2x+\lambda x^3=0,$$ which corresponds to the potential $\omega_0^2x^2/2+\lambda x^4/4$ in the Schrodinger equation. He derives two terms of the perturbative series of energy in powers of the Planck constant. Then he says:

This energy can be also determined using the Kramers-Born approach... The fact that one obtains exactly the same result seems to me to furnish remarkable support for the quantum-mechanical equations which have been taken here as a basis

My question: What exactly does he mean by Kramers-Born approach to anharmonic oscillator, what is the reference? (There are some references on Kramers and Born in the beginning of the paper, I looked at them, they do not seem to be relevant. Neither van-der-Waerden's commentary is helpful. Neither the book of Mehra and Rechenberg, Historical development of quantum theory helps.

So I repeat my question: What does Heisenberg compare his result with? This comparison makes him believe that his quantum mechanics is correct. This seems to be the crucial part of the paper.

• It may help attract answers if you would change the title to reflect that you're really asking specifically for the meaning of the phrase "Kramers-Born approach" as related to the anharmonic oscillators. Note also that concepts named using proper nouns can be very hard to track down because not only do the meanings change over time, but the original author may not have even used the phrase correctly in the first place. – DanielSank Oct 1 '15 at 19:38
• Dear @DanielSank Thanks for your comment: it shows that indeed someone read the question:-) I tried to reflect in the title EXACTLY what I want to know. Why this example, "anharmonic oscillator" was the main test of the new theory from Heisenberg's point of view. – Alexandre Eremenko Oct 1 '15 at 20:00
• If no one answers until the bounty expires, I may ask another question with a different title. – Alexandre Eremenko Oct 1 '15 at 20:01
• Well, I read it because it was in the featured questions list. If the title had indicated that I'd need to know what Kramers-Born meant to Heisenberg I wouldn't have even clicked. The point is that the title should attract people who might be able to answer. – DanielSank Oct 1 '15 at 20:41

Apparently Heisenberg referenced the perturbative approach to the quartic oscillator because his advisor and mentor, Max Born, was trying hard to use it in an attempt to push quantum theory past the Bohr model. Born actually invited Heisenberg to work on this problem in his group, and this is where Heisenberg had his breakthrough insight on the need for an entirely new conceptual framework. Born himself , on the other hand, tried to extend perturbation theory to non-integrable classical systems, in the idea of developing a sound framework for applying the phase coherences of the Bohr model beyond simple toy models. The reference to Born's exhaustive work is

M.Born, ”Vorlesungen ¨uber Atommechanik”, Springer, Berlin, 1925. English translation: ”The mechanics of the atom”, Ungar, New-York, 1927.

You can find all this in T.Paul's paper "On the status of perturbation theory", which gives a very nice outline of the history of it in Sec.4, "The birth of quantum mechanics".

Hope you enjoy it.

• Thanks for Born's book. I missed it somehow, and Sommerfeld's book which I read does not seem to address this point. – Alexandre Eremenko Oct 4 '15 at 1:52

Good answer by udrv. The importance of perturbative approach was also studied by Aitchison, McManus and Snyder. I think this work is one of the best that deals with Heisenberg's 'magical' paper.

As far as I can understand, there have been classical solutions for anharmonic oscillator by Born and Jordan. I don't know how Kramers is involved. The solution given by Heisenberg should correspond to classical solution in large quantum numbers. That may be the reason why Heisenberg cared to solved it.

• Thanks for the reference. I actually knew this paper, but perhaps have to look in it again. – Alexandre Eremenko Oct 4 '15 at 1:53