Quantum Mechanics
Problem 1
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?
Quantum Mechanics
Problem 2
Consider a one-dimensional step potential of the form:
⎩
⎨
⎧
=
0
0
)
(
V
x
V
0
0
≥
<
x
x
where
A quantum particle with mass m and energy
is incident on this
step “from the left” as shown in the figure.
.
0
0
>
V
0
V
E
>
(a)
Write down the appropriate solutions of the time-independent Schrödinger
equation for this particle in the x < 0 region and the x > 0 region.

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