*****We have shown that the Hermitian conjugate of a rotation operator is . Use this to prove that if the form an orthonormal complete set, then the set are also orthonormal and complete.- Given that is the one dimensional harmonic oscillator energy eigenstate:
a) Evaluate the matrix element
.
b) Write the upper left 5 by 5 part of the matrix.
- A spin 1 system is in the following state in the usual basis:
.
What is the probability that a measurement of the component of spin yields
zero?
What is the probability that a measurement of the component of spin
yields ?
- In a three state system, the matrix elements are given as
,
,
,
,
and
.
Assume all of the matrix elements are
real. What are the energy eigenvalues and eigenstates of the system?
At the system is in the state .
What is ?
- Find the (normalized) eigenvectors and eigenvalues of the (matrix) operator for
in the usual () basis.
*****A spin particle is in a magnetic field in the direction giving a Hamiltonian . Find the time development (matrix) operator in the usual basis. If , find .- A spin system is in the following state
in the usual basis:
.
What is the probability that a measurement of the component of spin yields
?
- A spin system is in the state
(in the usual eigenstate basis).
What is the probability that a measurement of yields
?
What is the probability that a measurement of yields
?
- A spin object is in an eigenstate of
with eigenvalue at t=0. The particle is in a magnetic
field = which makes the Hamiltonian for the system
. Find the probability to measure
as a function of time.
- Two degenerate eigenfunctions of the Hamiltonian are properly
normalized and have the following properties.

What are the properly normalized states that are eigenfunctions of H and P? What are their energies? - What are the eigenvectors and eigenvalues for the spin
operator ?
- A spin object is in an eigenstate of
with eigenvalue at t=0. The particle is in a magnetic
field = which makes the Hamiltonian for the system
. Find the probability to measure
as a function of time.
- A spin 1 system is in the following state, (in the usual
eigenstate basis):

What is the probability that a measurement of yields 0? What is the probability that a measurement of yields ? - A spin object is in an eigenstate of
with eigenvalue at t=0. The particle is in a magnetic
field = which makes the Hamiltonian for the system
. Find the probability to measure
as a function of time.
- A spin 1 particle is placed in an external field in the direction
such that the Hamiltonian is given by

Find the energy eigenstates and eigenvalues. - A (spin ) electron is in an eigenstate of
with eigenvalue
at . The particle is in a magnetic
field = which makes the Hamiltonian for the system
.
Find the spin state of the particle as a function of time.
Find the probability to measure
as a function of time.