Exercises

1. Demonstrate directly from the fundamental commutation relations for angular momentum, (300), that $[L^2, L_z] = 0$ , $[L^\pm, L_z] = \mp \,\hbar\,L^\pm$ , and $[L^+,L^-] = 2\,\hbar\,L_z$ .

2. Demonstrate from Equations (363)-(368) that

   $$L_x = {\rm i}\,\hbar\,\left(\sin\varphi\, \frac{\partial}{\partial \theta} + \cot\theta \cos\varphi\,\frac{\partial}{\partial \varphi}\right),$$

   $$L_y = -{\rm i} \,\hbar\,\left(\cos\varphi\, \frac{\partial}{\partial\theta} -\cot\theta \sin\varphi \,\frac{\partial}{\partial \varphi}\right),$$

   $$L_z = -{\rm i}\,\hbar\,\frac{\partial}{\partial\varphi},$$

   
   
   where $\theta$ , $\varphi$ are conventional spherical polar angles.

3. A system is in the state $\psi(\theta,\varphi)=Y_{l\,m}(\theta,\varphi)$ . Evaluate $\langle L_x\rangle$ , $\langle L_y\rangle$ , $\langle L_x^{\,2}\rangle$ , and $\langle L_y^{\,2}\rangle$ .

4. Derive Equations (385) and (386) from Equation (384).

5. Find the eigenvalues and eigenfunctions (in terms of the angles $\theta$ and $\varphi$ ) of $L_x$ .

6. Consider a beam of particles with $l=1$ . A measurement of $L_x$ yields the result $\hbar$ . What values will be obtained by a subsequent measurement of $L_z$ , and with what probabilities? Repeat the calculation for the cases in which the measurement of $L_x$ yields the results 0 and $-\hbar$ .

7. 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 $H$ ?

8. 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.)

9. 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.

10. 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. $$R_{2\,1}(r)= \frac{1}{\sqrt{3}\,(2\,a_0)^{3/2}}\,\frac{r}{a_0}\,\exp\left(-\frac{r}{2\,a_0}\right).$$