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# Lecture 16: Particle on a Ring ## Schrödinger equation $-\frac{\hbar^2}{2I}\frac{d^2}{d\phi^2}\Phi(\phi)=E\Phi(\phi)$ $I=mr^2$ = moment of inertia $\Phi(\phi+2\pi)=\Phi(\phi)$ ### General Solution $\Phi(\phi)=A e^{i m_l \phi} + B e^{-i m_l \phi}$ $-\frac{\hbar^2}{2I}\frac{d^2}{d\phi^2}\Phi(...

# Lecture 16: Particle on a Ring ## Schrödinger equation $-\frac{\hbar^2}{2I}\frac{d^2}{d\phi^2}\Phi(\phi)=E\Phi(\phi)$ $I=mr^2$ = moment of inertia $\Phi(\phi+2\pi)=\Phi(\phi)$ ### General Solution $\Phi(\phi)=A e^{i m_l \phi} + B e^{-i m_l \phi}$ $-\frac{\hbar^2}{2I}\frac{d^2}{d\phi^2}\Phi(\phi)=E\Phi(\phi)$ $-\frac{\hbar^2}{2I} (i m_l)^2 \Phi(\phi) = E\Phi(\phi)$ $\frac{\hbar^2 m_l^2}{2I} = E$ $E = \frac{\hbar^2 m_l^2}{2I} \quad m_l = 0, \pm 1, \pm 2,...$ ### Apply Boundary Condition $\Phi(\phi+2\pi) = \Phi(\phi)$ $A e^{i m_l (\phi+2\pi)} + B e^{-i m_l (\phi+2\pi)} = A e^{i m_l \phi} + B e^{-i m_l \phi}$ $e^{i m_l 2\pi} = 1$ $m_l = 0, \pm 1, \pm 2,...$ ### Normalize $\int_0^{2\pi} \Phi^* (\phi) \Phi(\phi) d\phi = 1$ $\Phi(\phi) = \frac{1}{\sqrt{2\pi}} e^{i m_l \phi}$ ### Summary $E = \frac{\hbar^2 m_l^2}{2I} \quad m_l = 0, \pm 1, \pm 2,...$ $\Phi(\phi) = \frac{1}{\sqrt{2\pi}} e^{i m_l \phi}$ ** двукратно вырожден (except for ml=0)** ### Angular Momentum $\hat{L_z} = -i\hbar \frac{\partial}{\partial \phi}$ $\hat{L_z} \Phi(\phi) = -i\hbar \frac{\partial}{\partial \phi} \frac{1}{\sqrt{2\pi}} e^{i m_l \phi}$ $\hat{L_z} \Phi(\phi) = m_l \hbar \Phi(\phi)$ ### Quantized $L_z = m_l \hbar$ ## Particle on a Sphere ### Schrödinger equation $\hat{H}Y(\theta, \phi)=EY(\theta, \phi)$ $\hat{H} = -\frac{\hbar^2}{2I} \Lambda^2 \qquad I = mr^2$ $\Lambda^2 = [\frac{1}{\sin\theta}\frac{\partial}{\partial \theta}(\sin\theta \frac{\partial}{\partial \theta}) + \frac{1}{\sin^2\theta}\frac{\partial^2}{\partial \phi^2}]$ ### Angular momentum $\hat{L}^2 = \hbar^2 \Lambda^2$ $\hat{L_z} = -i\hbar \frac{\partial}{\partial \phi}$ ### Eigenfunctions $Y(\theta, \phi)$ = Spherical Harmonics $Y_{l,m_l}(\theta, \phi)$ $l$ = azimuthal quantum number $m_l$ = magnetic quantum number ### Eigenvalues $\hat{L}^2 Y_{l,m_l}(\theta, \phi) = \hbar^2 l(l+1) Y_{l,m_l}(\theta, \phi) \qquad l = 0, 1, 2,...$ $\hat{L_z} Y_{l,m_l}(\theta, \phi) = \hbar m_l Y_{l,m_l}(\theta, \phi) \qquad m_l = -l, -l+1,..., 0,..., l-1, l$ ### Energy $E = \frac{\hbar^2 l(l+1)}{2I}$ ### Degeneracy $2l+1$