1.P Exercises
 According to classical physics, a nonrelativistic electron whose instantaneous acceleration is of magnitude radiates electromagnetic energy at the rate
 Demonstrate that
 Demonstrate that in a finite dimensional ket space: Here, , are general operators.

 If , are Hermitian operators then demonstrate that is only Hermitian provided and commute. In addition, show that is Hermitian, where is a positive integer.
 Let be a general operator. Show that , , and are Hermitian operators.
 Let be an Hermitian operator. Demonstrate that the Hermitian conjugate of the operator is .
 Let the be the eigenkets of an observable , whose corresponding eigenvalues, , are discrete. Demonstrate that
 Let the , where , and , be a set of degenerate eigenkets of some observable . Suppose that the are not mutually orthogonal. Demonstrate that a set of mutually orthogonal (but unnormalized) degenerate eigenkets, , for , can be constructed as follows:
 Demonstrate that the expectation value of a Hermitian operator is a real number. Show that the expectation value of an antihermitian operator is an imaginary number.
 Let be an Hermitian operator. Demonstrate that .
 Consider an Hermitian operator, , that has the property that , where is the unity operator. What are the eigenvalues of ? What are the eigenvalues if is not restricted to being Hermitian?
 Let be an observable whose eigenvalues, , lie in a continuous range. Let the , where
Contributors
 Richard Fitzpatrick (Professor of Physics, The University of Texas at Austin)