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$\newcommand{\ket}[1]{|#1\rangle}$When doing quantum mechanics on the circle $S^1$, it is well documented (yet seemingly controversial) that a self-adjoint "angle/position operator" $\hat{\phi}$ acting like $\hat{\phi} \ket{\phi} = \phi \ket{\phi}$ does not exist.

See:

That being said, I still see many people in recent papers using this approach. Am I missing something or is this the case of physicists' being defiant/uninformed/disregarding rigor? Perhaps there is a way to construct a non self-adjoint "angle-type" operator $\hat{\phi}$ (by taking limits or something etc) and this is what they mean?

An alternative approach instead quantizes via the two self-adjoint operators $\hat{c} = \widehat{\cos{\phi}}$ and $\hat{s} = \widehat{\sin{\phi}}$. Is this the only proper approach when doing quantum mechanics on the circle?

Qmechanic
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Eric Kubischta
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1 Answers1

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The rigorous way is to use $e^{i\phi}=c+is$ and the conjugate angular momentum operators, as the Hilbert space is spanned by single-valued functions on the circle. But physicists often get away with using $\phi$ but then remembering that physical results have to be $2\pi$ periodic in $\phi$.

Meng Cheng
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  • Are you saying physicists' define "$\hat{\phi} = -i \log( \hat{c} + i \hat{s})$" and then because the log has different branch cuts they have to choose one before this is well defined? – Eric Kubischta Dec 07 '21 at 18:37
  • I think that's right, it's often convenient to be able to directly work with $\phi$, but then it has to be restricted to $[0,2\pi)$, like a branch of the log as you said, and also one has to "remember" somehow that $\phi$ is only defined up to $2\pi$. – Meng Cheng Dec 07 '21 at 19:42