1
Problem 1
Do the onedimensional kinetic energy and momentum operators commute?
If not, what operator does their commutator represent?
ˆ
KE
=

¯
h
2
2
m
d
2
dx
2
ˆ
P
=

i
¯
h
d
dx
1.1
Solution
This question requires calculating the commutator of the operators given.
h
ˆ
KE,
ˆ
P
i
=

¯
h
2
2
m
d
2
dx
2

i
¯
h
d
dx


i
¯
h
d
dx

¯
h
2
2
m
d
2
dx
2
=
i
¯
h
3
2
m
d
2
dx
2
d
dx

d
dx
d
2
dx
2
=
i
¯
h
3
2
m
d
3
dx
3

d
3
dx
3
=
0
The operators commute; all done.
2
Problem 2
Given the following wavefunction, describing some quantum particle, ex
panded in a basis of eigenfunctions of a ”location” operator,
Ψ =
C
left
ψ
left
+
C
right
ψ
right
+
C
top
ψ
top
+
C
bottom
ψ
bottom
NOTE
:(the eigenfunction
ψ
left
is associated with an eigenvalue of ”LEFT”,
and so on for the other eigenfunctions)
2a. What is the probability of observing a value of ”TOP” (before any ex
plicit measurement is made).
If we have a detector that measures the ”LEFT” location, and a signal is
detected there after a particle is projected towards the array of detectors:
2a. How would you write the expression for the wavefunction?
2b. What is the probability of observing a value of ”LEFT”?
1
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2c.
If a measurement was made on the wavefunction 5 hours later, what
would be the probability of measuring a value of ”BOTTOM” (assume no
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 Fall '08
 Dybowski,C
 Physical chemistry, pH, Kinetic Energy, dx

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