4.P: Exercises

1. Demonstrate directly from the fundamental commutation relations for angular momentum, $\ref{300}$, that $[L^\pm, L_z] = \mp \,\hbar\,L^\pm$, and $L_x$
2. $L_y$ $L_z$ $\theta$, $\psi(\theta,\varphi)=Y_{l\,m}(\theta,\varphi)$. Evaluate $\langle L_y\rangle$, $\langle L_y^{\,2}\rangle$.
3. Derive Equations $\ref{385}$ and $\ref{386}$ from Equation $\ref{384}$.
4. Find the eigenvalues and eigenfunctions (in terms of the angles $\varphi$ ) of $l=1$. A measurement of $\hbar$. What values will be obtained by a subsequent measurement of $L_x$ yields the results 0 and $-\hbar$ .
5. The Hamiltonian for an axially symmetric rotator is given by $H = \frac{L_x^{\,2}+L_y^{\,2}}{2\,I_1} + \frac{L_z^{\,2}}{2\,I_2}.$ What are the eigenvalues of $H$ ?
6. The expectation value of $f({\bf x},{\bf p})$ in any stationary state is a constant. Calculate $0= \frac{d}{dt}\,(\langle{\bf x}\cdot{\bf p}\rangle) = \frac{\rm i}{\hbar}\,\langle[H, {\bf x}\cdot{\bf p}] \rangle$ for a Hamiltonian of the form $H = \frac{p^2}{2\,m} + V(r).$ Hence, show that $\left\langle\frac{p^2}{2\,m}\right\rangle = \frac{1}{2}\left\langle r\,\frac{dV}{dr}\right\rangle$ in a stationary state. This is another form of the Virial theorem. (See Exercise 8.)
7. Use the Virial theorem of the previous exercise to prove that $\left\langle \frac{1}{r}\right\rangle = \frac{1}{n^2\,a_0}$ for an energy eigenstate of the hydrogen atom.
8. Demonstrate that the first few properly normalized radial wavefunctions of the hydrogen atom take the form:
   1. $R_{1\,0}(r) = \frac{2}{a_0^{\,3/2}}\,\exp\left(-\frac{r}{a_0}\right).$
   2. $R_{2\,0}(r)= \frac{2}{(2\,a_0)^{3/2}}\left(1-\frac{r}{2\,a_0}\right)\exp\left(-\frac{r}{2\,a_0}\right).$
   3. $R_{2\,1}(r)= \frac{1}{\sqrt{3}\,(2\,a_0)^{3/2}}\,\frac{r}{a_0}\,\exp\left(-\frac{r}{2\,a_0}\right).$

Contributors

• Richard Fitzpatrick (Professor of Physics, The University of Texas at Austin)

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