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4.P: Exercises
 Last updated
 08:26, 16 Nov 2014

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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:


