Homework 3

Problem 1:

The Hamiltonian operator for a two state system is given by

H=a(|1><1|-|2><2|+|1><2|+|2><1|),

where a is a number with the dimensions of energy.

(a) Find the eigenvalues of H and the corresponding eigenkets |y₁> and |y₂> (as linear combinations of |1> and |2>).

(b) A unitary transformation maps the {|1>, |2>} basis onto the {|y₁>, |y₂>} basis. We have U|i>=|y_i>. Write down the matrix of U and the matrix of U^T in the {|1>, |2>} basis.

Problem 2:

Let P and Q be two linear operators and let [P,Q]=-ih. Find

(a)

(b)

(c)

Problem 3:

Suppose |i> and |j> are eigenkets of some Hermitian operator A. Under what conditions can we conclude that |i>+|j> is also an eigenket? Justify your answer.

Problem 4:

Using the rules of bra-ket algebra, prove or evaluate the following:

(a) tr(XY)=tr(YX), where X and Y are operators.
(b) (XY)^T=Y^TX^T
(c) exp[if(A)]=?, in ket-bra form, where A is a Hermitian operator whose eigenvalues are known.
(d) S_a'y*_a'(r')y_a'(r''), where y_a'(r')=<r'|a'>.

Problem 5:

Assume {|1>, |2>, |3>, ... , |n>} forms an orthonormal basis for the vector space V. Let W_ij be the matrix elements of the Hermitian operator W in this basis. Assume that the set {|1>, |2>, |3>, ... , |n>} is not an eigenbasis of W, but that the unitary transformation U|i>=|w_i> changes this basis to the eigenbasis {|w₁>, |w₂>, |w₃>, ... , |w_n>} of W. We have W|w_i>=w_i|w_i>. In this basis the matrix of W is diagonal. Let U_ij denote the matrix elements of the unitary operator in the {|1>, |2>, |3>, ... , |n>} basis, U_ij=<i|U|j>=<i|w_i>. The matrix elements of W in the {|w_i>} basis are the same as the matrix elements of U^TWU in the {|i>} basis. The matrix of U^TWU in the {|i>} basis is diagonal, <i|U^TWU|j>=0 if i¹j.

We can therefore diagonalize the matrix W by multiplying it from the left by U^T and from the right by U.
Consider the matrix in the {|i>} basis.

(a) Is this matrix Hermitian?

(b) Find its eigenvalues w_i and eigenvectors |w_i>.

(c) Find the matrix of U in the {|i>} basis.

(d) Find the matrix of U^T in the {|i>} basis. (U changes {|i>} into {|w_i>}.)

(e) Verify that UU^T=U^TU=I.

(f) Verify that the matrix U^TWU in the {|i>} basis is diagonal.