Monday, 25 December 2017

quantum mechanics - What periodic functions of the angle operator are Hermitian?



Let $\hat{\theta}$ be one of the position operators in cylindrical coordinates $(r,\theta,z)$. Then my question is, for what periodic functions $f$ (with period $2\pi$) is $f(\hat{\theta})$ a Hermitian operator? In case it's not clear, $f(\hat{\theta})$ is defined by a Taylor series.


The reason I ask is because of my answer here dealing with an Ehrenfest theorem relating angular displacement and angular momentum. This journal paper says that the two simplest functions which satisfy this condition are sine and cosine. First of all I'm not sure how to prove that they satisfy the condition, and second of all I want to see what other functions satisfy it.


What makes this tricky is that the adjoint of an infinite sum is not equal to the infinite sum of the adjoints.




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