Let
Ω be the operator defined bψ Ω = |Φ><ψ| where
|Φ>
and |ψ> are two vectors in a vector space V.

(a) Under what conditions is Ω Hermitian?

(b) Calculate Ω^{2}. Under what conditions is Ω a projector?

Solution:

- Concepts:

Mathematical foundations of quantum mechanics - Reasoning:

An operator A is Hermitian if A = A^{†}. A Hermitian operator satisfies <ψ|A|Φ> = <Φ|A|ψ>*.

A projector is a Hermitian operator. If Ω is a projector, then Ω^{2}= Ω. - Details of the calculation:

(a) Ω = |Φ><ψ|, Ω^{†}= |ψ><Φ|.

Ω = Ω^{†}--> Ω|ζ> = Ω^{†}|ζ> for any |ζ> in V.

We need |Φ><ψ|ζ> = |ψ><Φ|ζ>. This implies |Φ> = a|ψ>.

Then a|ψ><ψ|ζ> = |ψ><Φ|ζ> or a<ψ|ζ> = <Φ|ζ>.

But we also have <Φ|ζ> = a*<ψ|ζ>.

Therefore a = a*.

We need |Φ> = a|ψ>, with a real.

(b) Ω^{2}= |Φ><ψ|Φ><ψ|. We need <ψ|Φ> = 1 and |Φ> = a|ψ>.

This implies a<ψ|ψ> = 1, a = 1/<ψ|ψ>.

Quantum mechanics is often
conveniently formulated in terms of matrix operators. Let {|i>} be an
orthonormal basis for the state space.

(a) Prove that for any operator A we have ∑_{ij}|<i|A|j>|^{2}
= Tr(AA^{†}), where Tr denotes the trace.

(b) Derive the condition that must be satisfied for the product of two Hermitian operators to be itself a Hermitian operator.

(c) Prove that the trace of a matrix operator is invariant under a change of
representation, i.e. a change of basis.

Solution:

- Concepts:

Mathematical foundations of quantum mechanics - Reasoning:

This is a linear algebra problem. - Details of the calculation:

(a) ∑_{ij}|<i|A|j>|^{2}= ∑_{ij}<i|A|j><j|A^{†}|i> = ∑_{i}<i|AA^{†}|i> = Tr(AA^{†}), since ∑_{j}|j><j| = I.

(b) Let C = AB, where A and B are Hermitian operators.

For any |i>, |j> we have

<i|AB|j>* = <j|(AB)^{†}|i> = <j|B^{†}A^{†}|i> = <j|BA|i>.

If C is a Hermitian operator then <i|AB|j>* = <j|AB|i>, or

<j|BA|i> = <j|AB|i>. for any |i>, |j>.

C is a Hermitian operator implies that BA = AB, i.e. that A and B commute.

Let {|u>} be a second orthonormal basis for the state space.E.

Tr(A) = ∑_{i}<i|A|i> = ∑_{iuu'}<i|u><u|A|u'><u'|i> = ∑_{iuu'}<u'|i><i|u><u|A|u'>

= ∑_{uu'}<u'|u><u|A|u'> = ∑_{u'}<u'|A|u'> = Tr(A),

since ∑_{j}|j><j| = ∑_{u'}|u'><u'| = I.

Use the virial theorem for the eigenstates of atomic
hydrogen, namely, <nlm|T|nlm> = -E_{nl}, where E_{nl} is the eigenenergy of the |nlm> bound
state and T is the kinetic energy operator for an
electron bound to a proton, to evaluate the sum ∑_{nlm}|<nlm|**p**|n'l'm'>|^{2}, where
**p** is the
momentum operator and the sum is over a complete set of bound and continuum
states. Take the proton mass to be infinite and give your answer in terms of E_{n'l'}
and the mass m_{e} of the electron.

Solution:

- Concepts:

Mathematical foundations of quantum mechanics - Reasoning:

The eigenvectors of every Hermitian operator form a basis for the vector space,

∑_{nlm }| nlm ><nlm| = 1. - Details of the calculation:

∑_{nlm}|<nlm|**p**|n'l'm'>|^{2}= ∑_{nlm }<n'l'm'|**p**^{†}| nlm><nlm|**p**|n'l'm'>

= <n'l'm'|p^{2}|n'l'm'> = 2m_{e}<n'l'm'|T|n'l'm'> = -2m_{e}E_{n'l'}.

Note:**p**is a Hermitian operator,**p**=**p**^{T}.

Consider the matrix representation of the operator

(a) Is T Hermitian?

(b) Solve for the eigenvalues. Are they real?

(c) Determine the **normalized **eigenvectors. Since eigenvectors are not
unique to within a phase factor, arrange your eigenvectors so that the first
component of each is positive and real. Are they orthogonal?

(d) Using the eigenvectors as columns, construct U^{†}, the inverse of
the unitary matrix which diagonalizes T. Use this to find this diagonalized
version T^{d} = U^{†}TU.

What is special about the diagonal elements?

(e) Compare the determinant |T|, the trace Tr(T ), and eigenvalues of T to
those of T^{d}.

Solution:

- Concepts:

Matrix representation of operators. - Reasoning:

We are given the matrix of an operator in a particular basis. We are asked to find the eigenvalues of the operator and the eigenvectors of the operator in the given basis. We are then asked to make a unitary transformation to a basis in which the matrix of the operator is diagonal. - Details of the calculation:

(a) T_{ij}= T_{ji}*, T is Hermitian.

(b) eigenvalues α: (1 - α )(-α) - 2 = 0. α_{+}= 2, α_{-}= -1. The eigenvalues are real.

(c) eigenvector for α_{+}:

eigenvector for α_{-}:

The eigenvectors are orthogonal. <+|-> = <-|+> = 0.

(d)

U^{†}U = UU^{†}= I.

The diagonal values now are equal to the eigenvalues.

(e) det(T) = det(T^{d}) = -2, Tr(T) = Tr(T^{d}) = 1. Both matrices have the same eigenvalues.

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.

Solution:

- Concepts:

The eigenvalues and eigenvectors of a Hermitian operator. - Reasoning:

We are given enough information to construct the matrix of the Hermitian operator H in some basis. To find the eigenvalues E we set the determinant of the matrix (H - EI) equal to zero and solve for E. To find the corresponding eigenvectors {|Ψ>}, we substitute each eigenvalue E back into the equation (H-E*I)|Ψ> = 0 and solve for the expansion coefficients of |Ψ> in the given basis. - Details of the calculation:

(a) In the {|1>, |2>} basis the matrix of H is

.

The eigenvalues of H are found from

For the eigenvectors we find

(b) The matrix of U has the eigenvectors as its columns.

.