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Physics 401  Homework #8
1)
Threestate system (three points each).
Suppose that we have a physical system for
which there are only three states. For example, perhaps the system is a molecule where
the atoms can take on three different arrangements. Regardless of how we interpret the
physical meaning of the three states, we can write down the equations of quantum
mechanics for this system using these states as a basis. As usual, we will assume that the
states are orthonormal and complete. In other words, let's assume that any state can be
thought of as a superposition of the three base states, and that different states have no
overlap.
Let's further suppose that the Hamiltonian for this system in our threestate basis is:
3
0
0
0
0
0
0
ˆ
E
E
A
A
E
H
Then the timeindependent Schroedinger Equation is:
3
2
1
3
2
1
3
0
0
0
0
0
0
a
a
a
E
a
a
a
E
E
A
A
E
n
a) Find the three energy eigenvectors and energy eigenvalues (E
n
) for this system.
To answer this question, you may either use the methods of linear algebra, or you may
guess the eigenvectors, show that your guesses are correct, and find the eigenvalues
through direct substitution. (By the way, it's a good habit to always plug your
eigenvectors back into the original equation to check your work.)
b) Write down the fully timedependent solutions for the three stationary states of
this system. In other words, show how each eigenvector found in part (a) evolves in time.
c) We have not specified the physical meaning of our three base states, but there
is one question that we can answer about them. Are our base states the energy
eigenstates?
d) Sometimes people use Dirac notation to write down a Hamiltonian. If we label
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 Fall '11
 HALL
 Work, Quantum Physics

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