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# 4.P: Exercises

1. Demonstrate directly from the fundamental commutation relations for angular momentum, (300), that , , and .

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

where , are conventional spherical polar angles.

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

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

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

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

What are the eigenvalues of ?

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 8.)

9. Use the Virial theorem of the previous exercise to prove that

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.

2.

3.