Physics 751 Homework #3

Due Friday September 26, 11:00 am.

1. (a) If U and V are unitary, is UV unitary? Prove your result.

(b) If A and B are Hermitian, is AB Hermitian? Prove your result.

2. If H is Hermitian, prove

(a) U = e^iH is unitary,

(b) that log det U = iTrH.

3. Find the eigenvalues and eigenvectors of the Pauli matrices

Write down explicitly the unitary matrix that diagonalizes s_x. Can these two Hermitian matrices s_x , s_y be diagonalized simultaneously? Explain.

4. (a) Regarding use your result from problem 3 to write down its eigenvectors and eigenvalues.

(b) Find another form for U by expanding the exponential and summing the series to give well-known functions. (We shall be using this result later.)

5. Find the eigenvalues and eigenvectors of

and construct the unitary matrix which diagonalizes L_x.

6. Prove that both the determinant and the trace of a Hermitian matrix are unchanged in a unitary transformation, and hence find simple expressions for them in terms of the eigenvalues. (You may assume detAB = detA×detB.)