Name
Phys. 225/315
Feb. 29, 2008
MidTerm Exam 2
BOTH SIDES
75 points
Numerical answers may be left in terms of
π
,
√
2
,
μ
N
, etc.
Show all details of your calculations. Sketches and plots are very helpful.
1. (
15 pts
)
A twostate system has energy levels
E
a
and
E
b
with
E
a
< E
b
.
The system is prepared at
t
= 0 in a linear combination of the states

E
a
and

E
b
so
that there is a 25% probability that the energy is found to be
E
b
.
a) (
3 pts
)
Write the state of the system

ψ
(
t
)
at time
t
.

ψ
(
t
)
=
√
3
2

E
a
e

iE
a
t/
¯
h
+
1
2

E
b
e

iE
b
t/
¯
h
b) (
3 pts
)
Is there a time,
t
, when the system is certain to be found in state

E
a
?
Always
Never
Never: since
e

iE
b
t/
¯
h
is never zero.
c) (
3 pts
)
What is the first time
t
(for
t >
0) that there is a 25% probability
to find energy
E
b
?
Always
Never
Always: since the time dependence drops out in amplitude squared.
d) (
6 pts
)
Derive the expectation value of the Energy
E
, at time
t
.
Use
E
a

E
a
= 1 and
E
a

E
b
= 0
Find probability amplitude for each state
< E
a

ψ
(
t
) .
When you square this the time dependence drops out.
E
=
3
4
E
a
+
1
4
E
b
This is worked out in detail in McIntyre pages 51 and 52 equations 3.10 to 3.16
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2. (
15 pts
)
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 Spring '10
 ROTHBERG
 Energy, Magnetic Field, pts, Fundamental physics concepts, Nuclear magnetic resonance, precession frequency

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