Eigenvalues and Eigenfunctions of \( \hat{L}_z \)

In two-dimensional cylindrical coordinate system of \( (r,\phi) \), where \(r\) is the radial coordinate and \(\phi\) is the angular coordinate, the explicit form of the operator \( \hat{L}_z \) is \[ \hat{L}_z = -i\hbar \frac{\partial}{\partial \phi} \]

For eigenvalues and eigenfunctions of \( \hat{L}_z \):

\[ \hat{L}_z ~ \psi = l_z ~ \psi \]

Substituting for \( \hat{L}_z \), we get:

\[ -i\hbar \frac{\partial \psi}{\partial \phi} = l_z ~ \psi \]

To solve:

\[ \frac{\partial \psi}{\partial \phi} = i l_z \psi \]

Integrating with respect to \(\phi\):

\[ \psi(\phi) = e^{i l_z \phi} \]

Imposing periodic boundary condition:

\[ \psi(\phi + 2\pi) = \psi(\phi) \]

Leads to:

\[ e^{i m (\phi + 2\pi)} = e^{i m \phi} \]

So:

\[ e^{i m 2\pi} = 1 \]

Hence, \(m\) must be an integer: \(m = 0, \pm 1, \pm 2, \dots\) \( m \) is often called the magnetic quantum number.

Thus, the normalised eigenfunctions are \( \psi_m(\phi) = \frac{1}{\sqrt{}2\pi} e^{i m \phi} \) and the eigenvalues are \( l_z = m \hbar \).

Note the orthonormality relation \[ \langle \psi_m(\phi) | \psi_{m'}(\phi) \rangle = \delta_{m,m'}. \]