Heisenberg's Re-interpretation of Bohr-Sommerfeld Quantization Condition in his 1925 'Umdeutung' paper (p12)

[Rohin]

The paper I'm referring to can be accessed here: http://users.mat.unimi.it/users/galgani/arch/heis25ajp.pdf

On page 12, Werner Heisenberg, in the process of 're-interpreting atomic dynamics' from the principles of quantum theory, says that the Bohr Sommerfeld Quantization Rule, which says that the action integral of a period system over a closed region in the phase space is quantized and is equal to Planck's constant times a positive integer, appears arbitrary from the perspective of Bohr's Correspondence Principle?

Specifically, the equation he is referring to is equation (12) in the paper: loop_integral{pdq} = loop_integral{(x_dot)^2dt} = J(=nh), where n is an integer.

All I know is that the Correspondence Principle says that quantum systems in the limit of large quantum numbers should start emulating classical systems. I can't see how this points to a contradiction or an arbitrariness in the above quantization principle which Heisenberg seems to allude to in his paper. Could someone clarify this?

Thank you very much!

 

[gentzen]

Are you sure that it is on page 12? I couldn't find it there. And I am too lazy to ocr this document to make it searchable.

 

[Rohin]

Are you sure that it is on page 12? I couldn't find it there. And I am too lazy to ocr this document to make it searchable.

Correction: It's on page 7 of the document (page 267 of the article). My bad!

 

[pines-demon]

Read on correspondence principle here:

• https://plato.stanford.edu/entries/bohr-correspondence/

As with all Bohr principles, the guy was so cumbersome about it that we do not really use any of his principles anymore only Bohr understood them.

 

[gentzen]

the action integral of a period system over a closed region in the phase space is quantized and is equal to Planck's constant times a positive integer, appears arbitrary from the perspective of Bohr's Correspondence Principle?

Here is the relevant passage from the paper (bold emphasis by me):

In the earlier theory this phase integral was usually set equal to an integer multiple of h, i.e., equal to nh, but such a condition does not fit naturally into the dynamical calculation. It appears, even when regarded from the point of view adopted hitherto, arbitrary in the sense of the correspondence principle, because from this point of view the J are determined only up to an additive constant as multiples of h. ...
..., and in practice this indeterminacy has given rise to difficulties due to the occurence of half-integral quantum numbers.

Alfred Landé and Werner Heisenberg had defended such half-integral quantum numbers in 1921. This caused skepticism and critique. I would interpret this passage as a defence of that earlier work, hinting that it had always been correct, or at least not as wrong and misguided as some critics had suggested...