Qualifying Exam- Fall 2006
Electrodynamics
Problem 1
Twelve wires, each of resistance r, are connected to form the edges of a cube.
Calculate
the effective resistance R of this network across a body-diagonal of the cube.
Qualifying Exam 2006
Quantum Mechanics
Problem 2
Consider the three spin-1 matrices
⎟
⎟
⎟
⎠
⎞
⎜
⎜
⎜
⎝
⎛
=
0
1
0
1
0
1
0
1
0
2
h
x
S
;
⎟
⎟
⎟
⎠
⎞
⎜
⎜
⎜
⎝
⎛
−
−
=
0
0
0
0
0
2
i
i
i
i
S
y
h
;
⎟
⎟
⎟
⎠
⎞
⎜
⎜
⎜
⎝
⎛
−
=
1
0
0
0
0
0
0
0
1
h
z
S
(a)
Calculate the commutator of
and
.
x
S
y
S
(b)
What are the possible values we can get if we measure the spin along the x-axis?
(c)
Suppose we obtain the largest possible value when we measure the spin along the x-
axis.
If we now measure the spin along the z-axis, what are the probabilities for the
various outcomes?
Qualifying Exam- Fall 2006
Classical Mechanics (Escape Velocity)
Problem 3
Suppose the Moon were to have the same mass as the Earth, and you are trying to
throw one of your physics books from the Earth to the Moon.
With what minimum
velocity must the book leave the surface of the Earth?
Neglect the relative motion of the
Earth and them Moon, and the Earth’s rotation. The mass of the Earth is ME = 6.0·1024
kg, the radius of the Earth is 6.4 · 106 m, and the distance from the center of the Earth to
the center of the Moon is REM = 3.8 · 108 m. Compare your answer to the escape
velocity from Earth alone. The gravitational constant is G = 6.67 · 10
−
11 N m2/kg2.

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