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Consider the space of square integrable functions on a plane. \[ \iint dx\,dy |\psi(x,y)|^2 < \infty.\] Define radial and angular momenta operators \(\hat{p}_r, \hat{P}_\theta\) on the subset of functions satisfying \[\psi(r,\theta+2\pi) = \psi(r,\theta), \qquad \psi(r, \theta)|_{r=0} = \psi(r,\theta)|_{r\to \infty} =0 .\]
- Show that \({P}_{\theta}= -i\hbar\frac{\partial}{\partial\theta}\) satisfies \[\Big(\phi(r,\theta), \hat{P}_\theta \psi(r,\theta)\Big) = \Big(\hat{P}_\theta\phi(r,\theta), \psi(r,\theta)\Big) \] and is, therefore, a hermitian operator.
- Find the hermitian conjugate of the operator \(\hat{p}_r\equiv-i\hbar\frac{\partial}{\partial r}\). Show that \(\hat{p}_r\) is not a hermitian operator.
- Find a hermitian operator \(\hat{P}_r\) that may represent radial momentum in two dimensions.
- Consider the classical free Hamiltonian \( H_{cl} = \frac{P_r^2}{2m} + \frac{P_\theta^2}{2mr^2}.\) Replace the classical momenta \(P_r, P_\theta\) by corresponding hermitian momentum operators \(\hat{P}_r, \hat{P}_\theta\). Compare your answer for the operator so obtained with the free particle Schr\"{o}dinger Hamiltonian \[\widehat{H}_0 = -\frac{\hbar^2}{2m}\Big(\frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y}\Big).\] and give your comments.
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