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# Exercises

1. Demonstrate directly from the fundamental commutation relations for angular momentum, (4.11), that (a) , (b) , and (c) .

2. Demonstrate from Equations (4.74)-(4.79) that

where , are conventional spherical angles. In addition, show that

3. A system is in the state . Evaluate , , , and .

4. Derive Equations (4.108) and (4.109) from Equation (4.107).

5. Find the eigenvalues and eigenfunctions (in terms of the angles and ) of . Express the eigenfunctions in terms of the spherical harmonics.

6. Consider a beam of particles with . A measurement of yields the result . What values will be obtained by a subsequent measurement of , and with what probabilities? Repeat the calculation for the cases in which the measurement of yields the results 0 and .

7. The Hamiltonian for an axially symmetric rotator is given by

where and are the moments of inertia about the -axis (which corresponds to the symmetry axis), and about an axis lying in the - plane, respectively. What are the eigenvalues of ? [53]

8. The expectation value of in any stationary state is a constant. Calculate

for a Hamiltonian of the form

Hence, show that

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

for an energy eigenstate of the hydrogen atom whose principal quantum number is .

10. Suppose that a particle's Hamiltonian is

Show that and . [Hint: Use the Schrödinger representation.] Hence, deduce that

[Hint: Use the Heisenberg picture.] Demonstrate that if , where , then

11. Let

where is a non-negative integer. Show that

12. Demonstrate that the first few properly normalized radial wavefunctions of the hydrogen atom take the form:

13. Demonstrate that

for the hydrogen ground state. In addition, show that

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

15. Demonstrate that

where denotes a properly normalized energy eigenket of the hydrogen atom corresponding to the standard quantum numbers , , and .

16. Let denote the expectation value of for an energy eigenstate of the hydrogen atom characterized by the standard quantum numbers , , and .
1. Demonstrate that

where

and is a well-behaved solution of the differential equation

2. Integrating by parts, show that

and

as well as

3. Demonstrate from the governing differential equation for that

4. Combine the final result of part (b) with the governing differential equation to prove that

5. Combine the results of parts (c) and (d) to show that

Hence, derive Kramers' relation:

6. Use Kramers' relation to prove that

17. Let , where is a properly normalized radial hydrogen wavefunction corresponding to the conventional quantum numbers and , and is the Bohr radius.
1. Demonstrate that

2. Show that in the limit .
3. Demonstrate that

4. Hence, deduce that

for .

Next: Spin Angular Momentum Up: Orbital Angular Momentum Previous: Energy Levels of Hydrogen
Richard Fitzpatrick 2016-01-22