I'm being asked on a homework to show that the m orbital angular momentum quantum number can only take integer values. Using ladder operators I know how to prove that m is restricted to half-integers, but I'm having trouble with a further restriction. I'm quite certain the problem does not want me to generally solve Laplace's Equation for spherical harmonics.(adsbygoogle = window.adsbygoogle || []).push({});

The method which I've seen but has trouble convincing me is the following:

[tex]

-i \hbar \frac{\partial \Phi}{\partial \phi^2} \ = \ m \ \hbar \ \Phi

[/tex]

[tex]

\Phi \ = \ e^{im \phi}

[/tex]

Now this requirement:

[tex]

\Phi(\phi + 2 \pi) \ = \ \Phi(\phi)

[/tex]

Producing the desired quantization. This feels a little artificial though - on one hand it seems obvious that the wavefunction should be single-valued at a point, the addition of a phase factor wouldn't change any predictions, would it? Any calculations of an observable for a point would be the same under a full rotation, unless I'm missing something. Is there a way to either make this more concrete or a better way to show this quantization?

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# Homework Help: Question about quantization of Lz

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