Operators(1)

Operators

Problem:
Quantum mechanics is often conveniently formulated in terms of matrix operators. Let {|i>} be an orthonormal basis for the state space E.
(a) Prove that for any operator A,

, 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, principles, relations that apply to the problem:
This is a linear algebra problem.

Why do they apply?
We are asked to proof some theorems from linear algebra.

How do they apply?

(a) ,

since .

Details of the calculations:
(b) Let C=AB, where A and B are Hermitian operators. For any |i>, |j> we have

If C is a Hermitian operator, then , or for any |i>, |j>. C=AB is Hermitian implies that BA=AB, i.e. A and B commute.

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

Problem:
Consider a one-dimensional problem. Let the translation operator T(a) describe the operation T(a)y(x)=y(x+a), where a is a constant displacement.
(a) Show that this operator commutes with the Hamiltonian

, if the potential has the periodic property V(x)=V(x+a).
(b) Let y(x) be an eigenstate of T(a) with eigenvalue c. Show that c is a constant of motion

Solution:

Concepts, principles, relations that apply to the problem:
Commuting observables, the evolution of the mean value of an observable.

Why do they apply?
To show that T(a) commutes with H we show that for any y(x) we have as long as V(x+a)=V(x). To do this, we must express T(a) as an operator in coordinate space. If an operator commutes with H, and does not contain time explicitly, then it is a constant of motion.

How do they apply?

(a) , (Taylor series expansion).

For any y(x) we have since V(x+a)=V(x). Thus [T(a),H]=0.

Details of the calculations:

(b) T(a)y(x)=cy(x). Since T(a) does not explicitly depend on time we have . Since y(x) is an eigenstate of T(a) we have <T(a)>=c. Therefore c is a constant of motion.
Think about: y(x) represents the projection of |y> onto the basis vector |x>. What does T(a) do to the basis vectors {|x>} ?

Problem:
Given are the Hamiltonian operator

for the one-dimensional Schroedinger equation and the canonical commutation relations [p,x]=

.
(a) Prove the following identity: [x,[H,x]]=

(b) Let |y_n> denote the eigenstates of the Hamiltonian H|y_n>=E_n|y_n>. Take the ground-state matrix element of part (a) and use the completeness relation to establish the following sum rule: . Note that this is an exact result.

(d) Discuss how this relation is satisfied in the one-dimensional simple harmonic oscillator.

Solution:

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Problem:

Consider a two dimensional vector space with an orthonormal basis {|1>, |2>}. Consider an operator whose matrix in that basis is .

(a) Show that the s_y operator is Hermitian?
(b) Calculate its eigenvalues and eigenvectors.
(c) Calculate the matrices which represent the projectors onto these eigenvectors. Verify that they satisfy the orthogonality and completeness relation.

Solution:

(a) is Hermitian. It is the matrix of a Hermitian operator. Its eigenvalue are real.

(b) To find the eigenvalues we set .

The eigenvector associated with is with .

matrix of P₁: , matrix of P₂:

Problem:
Consider the Hermitian Hamiltonian H=H₀+H’, where H’ is a small perturbation. Assume that the exact solutions H₀|y>=E₀|y> are known, that there are two of them, and that they are orthogonal and degenerate in energy. Derive from first principles an expression to first order in H’ for the energies of the perturbed levels in terms of the matrix elements of H’.

Solution:

Concepts, principles, relations that apply to the problem:
Two-state systems.

Why do they apply?
We are asked to find the energy levels of a two state system with Hamiltonian H=H₀+H', where H₀|y >=E₀|y >, E₀ being two-fold degenerate.

How do they apply?

The matrix of H’ is .

The matrix of H is .

These are the exact eigenvalues of H. In the case of a two level system with a degenerate Hamiltonian H₀ first order perturbation theory requires finding the eigenvalues of H’ and adding them to E₀, which also gives the exact result.