Exercises

1. Demonstrate directly from the fundamental commutation relations for angular momentum, (4.11), that (a) $[L^2, L_z] = 0$ , (b) $[L^\pm, L_z] = \mp \,\hbar\,L^\pm$ , and (c) $[L^+,L^-] = 2\,\hbar\,L_z$ .

2. Demonstrate from Equations (4.74)-(4.79) 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 angles. In addition, show that

   $$L^\pm = \pm \hbar\,\exp(\pm{\rm i}\,\varphi)\left(\frac{\partial}{\partial\theta} \pm{\rm i} \,\cot\theta\,\frac{\partial}{\partial\varphi}\right),$$

   $$L^2 = - \hbar^{\,2}\left( \frac{1}{\sin\theta}\frac{\partial}{\partia... ...} + \frac{1}{\sin^2\theta}\frac{\partial^{\,2}} {\partial\varphi^{\,2}}\right).$$

   
   

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 (4.108) and (4.109) from Equation (4.107).

5. Find the eigenvalues and eigenfunctions (in terms of the angles $\theta$ and $\varphi$ ) of $L_x$ . Express the $l=1$ eigenfunctions in terms of the spherical harmonics.

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
   $$H = \frac{L_x^{\,2}+L_y^{\,2}}{2\,I_\perp} + \frac{L_z^{\,2}}{2\,I_\parallel},$$
   where $I_\parallel$ and $I_\perp$ are the moments of inertia about the $z$ -axis (which corresponds to the symmetry axis), and about an axis lying in the $x$ -$y$ plane, respectively. What are the eigenvalues of $H$ ? [53]

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 9.) [53]

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 whose principal quantum number is $n$ .

10. Suppose that a particle's Hamiltonian is
    $$H = \frac{p^{\,2}}{2\,m} + V({\bf x}).$$
    Show that $[{\bf L},p^{\,2}] = {\bf0}$ and $[{\bf L},V({\bf x})]=-{\rm i}\,\hbar\,{\bf x}\times \nabla V$ . [Hint: Use the Schrödinger representation.] Hence, deduce that
    $$\frac{d\langle {\bf L}\rangle}{dt} = -\langle {\bf x}\times \nabla V\rangle.$$
    [Hint: Use the Heisenberg picture.] Demonstrate that if $V=V(r)$ , where $r=\vert{\bf x}\vert$ , then
    $$\frac{d\langle {\bf L}\rangle}{dt} ={\bf0}.$$

11. Let
    $$I_n = \int_0^\infty y^{\,n}\,e^{-y},$$
    where $n$ is a non-negative integer. Show that
    $$I_n = n!.$$

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

    4. $$R_{3\,0}(r)= \frac{2}{(3\,a_0)^{3/2}}\left(1-\frac{2\,r}{3\,a_0}+\frac{2\,r^{\,2}}{27\,a_0^{\,2}}\right)\exp\left(-\frac{r}{3\,a_0}\right).$$

    5. $$R_{3\,1}(r)= \frac{4\sqrt{2}}{9\,(3\,a_0)^{3/2}}\,\frac{r}{a_0}\left(1-\frac{r}{6\,a_0}\right)\exp\left(-\frac{r}{3\,a_0}\right).$$

    6. $$R_{3\,2}(r)= \frac{2\sqrt{2}}{27\sqrt{5}\,(3\,a_0)^{3/2}}\left(\frac{r}{a_0}\right)^2\exp\left(-\frac{r}{3\,a_0}\right).$$

13. Demonstrate that
    $$\langle r^{\,k}\rangle = \frac{(2+k)!}{2^{1+k}}\,a_0^{\,k}$$
    for the hydrogen ground state. In addition, show that
    $$\langle x^{\,2}\rangle =\langle y^{\,2}\rangle =\langle z^{\,2}\rangle = a_0^{\,2}.$$

14. Show that the most probable value of $r$ in the hydrogen ground state is $a_0$ .

15. Demonstrate that
    $$\langle 2,0,0\vert\,z\,\vert 2,1,0\rangle = -3\,a_0,$$
    where $\vert n,l,m\rangle$ denotes a properly normalized energy eigenket of the hydrogen atom corresponding to the standard quantum numbers $n$ , $l$ , and $m$ .

16. Let $\langle r^{\,k}\rangle$ denote the expectation value of $r^{\,k}$ for an energy eigenstate of the hydrogen atom characterized by the standard quantum numbers $n$ , $l$ , and $m$ .
    1. Demonstrate that
       $$\langle r^{\,k}\rangle = (n\,a_0)^{\,k+3}\,J_k,$$
       where
       $$J_k = \int_0^\infty dx\,x^{\,k}\,[u(x)]^{\,2},$$
       and $u(x)$ is a well-behaved solution of the differential equation
       $$u'' = \left[\frac{l\,(l+1)}{x^{\,2}}-\frac{2\,n}{x}+1\right] u.$$

    2. Integrating by parts, show that
       $$\int_0^\infty dx\, x^{\,k}\,u\,u' =-\frac{k}{2}\,J_{k-1},$$
       and
       $$\int_0^\infty dx\,x^{\,k}\,(u')^{\,2} = -\frac{2}{k+1}\int_0^\infty dx\,x^{\,k+1}\,u'\,u'',$$
       as well as
       $$\int_0^\infty dx\,x^{\,k}\,u\,u'' = \frac{k\,(k-1)}{2}\,J_{k-2} +\frac{2}{k+1}\,\int_0^\infty dx\,x^{\,k+1}\,u'\,u''.$$

    3. Demonstrate from the governing differential equation for $u(x)$ that
       $$\int_0^\infty dx\,x^{\,k}\,u\,u'' = l\,(l+1)\,J_{k-2}-2\,n\,J_{k-1}+J_k.$$

    4. Combine the final result of part (b) with the governing differential equation to prove that
       $$\int_0^\infty dx\,x^{\,k}\,u\,u'' =-\frac{l\,(l+1)\,(k-1)}{k+1}\,J_{k-2} +\frac{2\,n\,k}{k+1}\,J_{k-1} -J_k +\frac{k\,(k-1)}{2}\,J_{k-2}.$$

    5. Combine the results of parts (c) and (d) to show that
       $$\frac{k}{4}\,[(2\,l+1)^2-k^{\,2}]\,J_{k-2} -n\,(2\,k+1)\,J_{k-1}+(k+1)\,J_k = 0.$$
       Hence, derive Kramers' relation:
       $$\frac{k\,a_0^{\,2}}{4}\,[(2\,l+1)^2-k^{\,2}]\,\langle r^{\,k-2}\r... ...\,\langle r^{\,k-1}\rangle +\frac{(k+1)}{n^{\,2}}\,\langle r^{\,k}\rangle = 0.$$

    6. Use Kramers' relation to prove that

       $$\left\langle \frac{1}{r}\right\rangle = \frac{1}{n^{\,2}\,a_0},$$

       $$\langle r\rangle = \frac{a_0}{2}\,[3\,n^{\,2}-l\,(l+1)],$$

       $$\langle r^{\,2}\rangle = \frac{a_0^{\,2}\,n^{\,2}}{2}\,[5\,n^{\,2}+1-3\,l\,(l+1)].$$

       
       

17. Let $R_{nl}(r)= v_{nl}(r/a_0)/(r/a_0)$ , where $R_{nl}(r)$ is a properly normalized radial hydrogen wavefunction corresponding to the conventional quantum numbers $n$ and $l$ , and $a_0$ is the Bohr radius.
    1. Demonstrate that
       $$\frac{d^{\,2}v_{nl}}{dy^{\,2}} = \left[\frac{l\,(l+1)}{y^{\,2}} -\frac{2}{y} + \frac{1}{n^{\,2}}\right]v_{nl}.$$

    2. Show that $v_{nl}\sim y^{\,1+l}$ in the limit $y\rightarrow 0$ .
    3. Demonstrate that
       $$\left(\frac{1}{n^{\,2}}-\frac{1}{m^{\,2}}\right)\int_0^\infty dy\,v_{nl}(y)\,v_{ml}(y) = 0.$$

    4. Hence, deduce that
       $$\int_0^\infty dr\, r^{\,2}\,R_{nl}(r)\,R_{ml}(r)=0$$
       for $n\neq m$ .