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Physics 401  Homework #5
1)
A quantum mechanical state in two bases.
A quantum mechanical particleinabox
is in a superposition of two stationary states, the n = 4 state and the n = 5 state. The
superposition is an equal mixture of the two states, but the n = 5 state is outofphase by
an angle of
compared to the n = 4 state. We can represent this state by a ketvector:
state
QM
favorite
our
,
but this symbol doesn't tell us much about the state. We can, however, write the state in
other forms which are more explicit, and more useful for doing calculations. For
example, we can project the state into the position basis, or we can project it into the
energy basis.
a) (3 points) Write down an explicit representation of this state in the position (x)
basis. In other words, write down the wave function
(x) for this state.
b) (3 points) Write down an explicit representation for this state in the energy
basis. In other words, write down the {a
n
} for this state in a column vector format.
c) (3 points) What is the energy basis good for? To answer this, name one type of
question that requires almost no calculation to answer in the energy basis. Also name a
second type of question that is similarly easy to answer in the position basis.
d) (3 points) Calculate the overlap between this state and the state from
Homework #4, question 1, working in the energy basis. In other words, calculate
state
QM
favorite
our
question
Homework
for
state
QM
the
1
4
using the columnvector and rowvector format of the energy basis. You may use the
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 Fall '11
 HALL
 Work, Quantum Physics

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