4.P: Exercises
 Demonstrate directly from the fundamental commutation relations for angular momentum, (300), that , , and .
 Demonstrate from Equations (363)(368) that
 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
 The expectation value of in any stationary state is a constant. Calculate
 Use the Virial theorem of the previous exercise to prove that
 Demonstrate that the first few properly normalized radial wavefunctions of the hydrogen atom take the form:

Contributors
 Richard Fitzpatrick (Professor of Physics, The University of Texas at Austin)