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- Demonstrate directly from the fundamental commutation relations for angular momentum, (300), that , , and .
- Demonstrate from Equations (363)-(368) that
where , are conventional spherical polar angles.
- A system is in the state . Evaluate , , , and .
- Derive Equations (385) and (386) from Equation (384).
- Find the eigenvalues and eigenfunctions (in terms of the angles and ) of .
- 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 .
- The Hamiltonian for an axially symmetric rotator is given by
What are the eigenvalues of ?
- 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.)
- Use the Virial theorem of the previous exercise to prove that
for an energy eigenstate of the hydrogen atom.
- Demonstrate that the first few properly normalized radial wavefunctions of the hydrogen atom take the form: