Consider a quantum mechanical system with state spaceH=C^3 and Hamil-
tonian operator
H=
0 1 0
1 0 0
0 0 2
Solve the Schr ̈odinger equation for this system to find its state vector|Ψ(t)〉
at any timet >0, given that the state vector att= 0 was
ψ 1
ψ 2
ψ 3
withψi∈C.
B.2 Chapters 3 and
Problem 1:
Calculate the exponentialetMfor
0 π 0
−π 0 0
0 0 0
by two different methods:
- Diagonalize the matrixM(i.e., write asPDP−^1 , forDdiagonal), then
show that
etPDP
− 1
=PetDP−^1
and use this to computeetM.- CalculateetMusing the Taylor series expansion for the exponential, as
well as the series expansions for the sine and cosine.
Problem 2:
Consider a two-state quantum system, with Hamiltonian
H=−Bxσ 1(this is the Hamiltonian for a spin^12 system subjected to a magnetic field in the
x-direction).
- Find the eigenvectors and eigenvalues ofH. What are the possible energies
that can occur in this quantum system? - If the system starts out at timet= 0 in the state
|ψ(0)〉=(
1
0
)
(i.e., spin “up”) find the state at later times.