量子力学作业2009

Phys500 Problem set 1

1.For the Pauli matrices , prove

a.

b. (for

c.

d.Tr()=0

e.det(

2. that commute with

3.Show that if any operator commutes with two components of an angular momentum, it

commutes with all three.

4.Show that if is a Hermitian operator, then is a unitary operator.

5.The Hamiltonian for a particle of mass m moving in a quadratic potential is

where

ing , find the expectation

values

b.Solve for the first two eigenfunctions in the position representation.

c.Solve for the first two eigenfunctions in the momentum representation.

6.

a.What is the state of the system at time t?

b.What is the probability the system is in eigenstate |2> at time t?

c.Find the expectation values for the state at time t, , if we have

initial conditions

7.Heisenberg representation: Define the unitary operator , where H is the

Hamiltonian operator of the system.

a.Show that generates the time evolution of any state:

b.Define the time-dependent operators .

Show that

and .

This is the Heisenberg representation, in which one specifies the (constant) initial state of the system , and then quantities of interest are found by evaluating time-dependent operators acting on this constant state, as opposed to having constant operators acting on time-dependent states.

c.Show that for the SHO,

d.Check that you get the same answer for the expectation values as in question 6.c.

8.Consider a system with a 3-dimensional state space. Let the operator O be represented by

a.Find the eigenvectors and eigenvalues of O

b.. What is the probability of measuring each of the

eigenvalues?

9. A deuterium atom is composed of a nucleus of spin I=1 and an electron. The electron angular

momentum is where is its orbital angular momentum and is its spin. The total angular momentum of the atom is

a.What are the possible values for the quantum numbers J and F for a deuterium atom in

the ground (1s) state?

b.What are the values if the atom is in the excited state?

10.Let be the total angular momentum of a system consisting of 3 spin ½

particles, with no orbital angular momentum. By first adding two, then the third to the result, find the basis of eigenvectors of the total spin operators .

11.The Hamiltonian for a spin-1/2 particle in a magnetic field is where the magnetic

moment is .

a.Find the eigenvectors and eigenvalues of this Hamiltonian. You may as well take the

magnetic field to be in the z direction.

b. at time t=0. Find the

probability that the particle is in state with as a function of t.