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 |ψ_{1}> and
|ψ_{2}> (as linear combinations of |1> and |2>).

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

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

(a) [Q,P],

(b) [Q,P^{n}],

(c) [P,Q^{n}] .

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.

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)^{† }= Y^{†}X^{†}

(c) exp(i f(A)) = ?, in ket-bra form, where A is a Hermitian operator
whose eigenvalues are known.

(d) ∑_{a'}ψ*_{a'}(**r**')ψ_{a'}(**r**''),
where ψ_{a'}(**r**') = <**r**'|a'>.

Assume {|1>, |2>, |3>, ... , |n>} forms an orthonormal basis for the
vector space V. Let *Ω*_{ij} be the matrix
elements of the Hermitian operator *Ω* in this basis.
Assume that the set {|1>, |2>, |3>, ... , |n>} is not an
eigenbasis of *Ω*, but that the unitary transformation
U|i> = |ω_{i}>
changes this basis to the eigenbasis {|ω_{1}>, |ω_{2}>, |ω_{3}>,
... , |ω_{n}>} of *Ω*.
We have *Ω*|ω_{i}> = ω_{i}|ω_{i}>.
In
this basis the matrix of *Ω* 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|ω_{i}>.
The
matrix elements of *Ω* in the {|ω_{i}>}
basis are the same as the matrix elements of U^{†}ΩU
in the {|i>} basis. The matrix of U^{†}ΩU
in the {|i>} basis is diagonal, <i|U^{†}ΩU|j> = 0
if i ≠ j.

<i|U^{†}ΩU|j> = ∑_{kl}<i|U^{†}|k><k|Ω|l><l|U|j>
= ∑_{kl}U^{†}_{ik}Ω_{kl}U_{lj}.

We can therefore diagonalize the matrix
*Ω* by
multiplying it from the left by U^{†} and from the right by
U.

Consider the matrix

in the {|i>}
basis.

(a) Is this matrix Hermitian?

(b) Find its eigenvalues ω_{i} and eigenvectors |ω_{i}>.

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

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

(e) Verify that UU^{† }= U^{†}U = I.

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