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# Lecture 18: Particle on a Ring ## Schrödinger equation $-\frac{\hbar^2}{2I}\frac{d^2}{d\phi^2}\Theta(\phi) = E\Theta(\phi)$ $\Theta(\phi) = A e^{im\phi}$ $m = 0, \pm 1, \pm 2,...$ $E = \frac{\hbar^2 m^2}{2I}$ ## Wave Function $\Theta_m(\phi) = \frac{1}{\sqrt{2\pi}}e^{im\phi}$ ## Probabilit...

# Lecture 18: Particle on a Ring ## Schrödinger equation $-\frac{\hbar^2}{2I}\frac{d^2}{d\phi^2}\Theta(\phi) = E\Theta(\phi)$ $\Theta(\phi) = A e^{im\phi}$ $m = 0, \pm 1, \pm 2,...$ $E = \frac{\hbar^2 m^2}{2I}$ ## Wave Function $\Theta_m(\phi) = \frac{1}{\sqrt{2\pi}}e^{im\phi}$ ## Probability Density $\begin{aligned} |\Theta_m(\phi)|^2 &= \Theta_m^*(\phi)\Theta_m(\phi) \\ &= \frac{1}{\sqrt{2\pi}}e^{-im\phi}\frac{1}{\sqrt{2\pi}}e^{im\phi} \\ &= \frac{1}{2\pi} \end{aligned}$ ## Key Points * $E_m \propto m^2$ * $E_m \geq 0$ * $E_m = E_{-m}$ * $\Delta E_{m+1,m} \propto 2m+1$ ## Operators $\hat{L}_z = -i\hbar\frac{\partial}{\partial \phi}$ $\hat{L}_z\Theta_m(\phi) = -i\hbar\frac{\partial}{\partial \phi}\frac{1}{\sqrt{2\pi}}e^{im\phi} = m\hbar\frac{1}{\sqrt{2\pi}}e^{im\phi} = m\hbar\Theta_m(\phi)$ $\hat{L}_z\Theta_m(\phi) = m\hbar\Theta_m(\phi)$ $\langle L_z\rangle = \int_0^{2\pi}\Theta_m^*(\phi)\hat{L}_z\Theta_m(\phi)d\phi = m\hbar\int_0^{2\pi}\Theta_m^*(\phi)\Theta_m(\phi)d\phi = m\hbar$ $\langle L_z^2\rangle = \int_0^{2\pi}\Theta_m^*(\phi)\hat{L}_z^2\Theta_m(\phi)d\phi = m^2\hbar^2\int_0^{2\pi}\Theta_m^*(\phi)\Theta_m(\phi)d\phi = m^2\hbar^2$ $\sigma_{L_z}^2 = \langle L_z^2\rangle - \langle L_z\rangle^2 = m^2\hbar^2 - (m\hbar)^2 = 0$