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Page 1
Family Name
(Please
print
)
Given Name(s)
Student Number
Tutor
PHY294S — 2012
TEST I (Quantum Physics)
16 February 2012
Duration: One hour
Aids allowed:
Type 3 calculator (non-programmable, non-graphic and without alphanumeric storage).
Before starting, please print your name, student number and tutor’s name at the top of this page and on the test
booklet.
This test has three questions. The first one is worth 10 marks, and the other two are each worth 20 marks. Answer all
questions.
Do not separate the three stapled sheets of this question paper. At the end of the test, put the question paper inside
your answer booklet before handing it to the invigilator.
Good luck!
Possibly useful equations
Speed of light
c
=3
.
00
×
10
8
m/s
Mass of electron
m
e
=9
.
11
×
10
-
31
kg
= 511
keV/c
2
Elementary charge
e
=1
.
602
×
10
-
19
C
Mass of proton
m
p
=1
.
67
×
10
-
27
kg
= 939
MeV/c
2
Coulomb constant
k
e
=8
.
99
×
10
9
Jm
/
C
2
Planck’s constant
h
=6
.
626
×
10
-
34
Js
=4
.
14
×
10
-
15
eV s
hc
=1
.
240
keV nm
1
eV
=1
.
602
×
10
-
19
J
=
h/
2
π
E
=
=
ω
λ
dB
=
h
p
p
=
h
λ
=
k
-
2
2
m
2
Ψ(
r
,t
)+
U
(
r
,t
) Ψ(
r
,t
)=
H
Ψ(
r
,t
)=
i
∂t
Ψ(
r
,t
)
2
ψ
(
r
)=
2
m
2
(
U
(
r
)
-
E
)
ψ
(
r
)
Ψ(
r
,t
)
free
= cos(
k
·
r
-
ωt
)+
i
sin(
k
·
r
-
ωt
)=
A
e
i
(
k
·
r
-
ωt
)
=
A
e
i
(
p
·
r
-
Et
)
/
all space
Ψ(
r
,t
)
2
d
3
r
=1
Ψ
k
(
r
,t
)=
ψ
k
(
r
)
e
-
i
E
k
t/
for
any
U
(
r
)
Ψ
n
(
x, t
)=
1
a
cos
nπx
2
a
e
-
i
E
n
t
(
n
odd
)
-
a
x
a
1
a
sin
nπx
2
a
e
-
i
E
n
t
(
n
even
)
-
a
x
a
0
x<
-
a
or
x>a
E
n
=
2
k
2
2
m
=
2
π
2
n
2
8
ma
2
ψ
(
x
)=
A
1
e
i
k
1
x
+
B
1
e
-
i
k
1
x
k
1
=
2
m
(
E
-
U
0
)
2
(
E>U
0
=
constant
)
A
2
e
k
2
x
+
B
2
e
-
k
2
x
k
2
=
2
m
(
U
0
-
E
)
2
(
E<U
0
=
constant
)
R
=
|
B
1
|
2
|
A
1
|
2
T
=
k
t
|
A
t
|
2
k
i
|
A
1
|
2
all space
Ψ
*
m
(
r
,t
n
(
r
,t
)d
3
r
=
δ
mn


Page 2
Φ(
r
,t
)=
k
B
k
Ψ
k
(
r
,t
)
=
k
B
k
2
=1
A
=
all space
Φ
*
(
r
,t
)
A
Φ(
r
,t
)
d
3
r
Observable
Operator
Eigenfunction
Eigenvalue
Position
r
δ
(
r
-
r
0
)
r
0
Momentum
-
i
e
i
p
·
r
/
p
Energy
i
t
e
-
i
Et/
E
A
Ψ
n
(
r
,t
)=
a
n
Ψ
n
(
r
,t
)=
⇒
A
=
k
|
B
k
|
2
a
k
Δ
A
=
A
2
-
A
2
E
n
=
p
2
n
2
m
+
U
(
r
)
n
U
SHO
=
1
2
Kx
2
=
1
2
2
x
2
s
=
x
E
n
=
n
+
1
2
ω
ψ
0
(
s
)=
1
π
1
/
4
e
-
s
2
/
2
ψ
1
(
s
)=
2
π
s
e
-
s
2
/
2
ψ
2
(
s
)=
1
2
π
(2
s
2
-
1)
e
-
s
2
/
2
U
(
x
)
U
(
x
0
)+
1
2
d
2
U
(
x
)
d
x
2
x
0
(
x
-
x
0
)
2
z
=
R
e
i
θ
=
R
cos
θ
+
i
R
sin
θ
=
a
+
i
b
a
=
R
cos
θ
R
=
a
2
+
b
2
b
=
R
sin
θ
θ
= tan
-
1
(
b/a
)
cos
θ
=
e
i
θ
+
e
-
i
θ
2
sin
θ
=
e
i
θ
-
e
-
i
θ
2
i
x
cos(
ax
)d
x
=
cos(
ax
)
a
2
+
x
sin(
ax
)
a
+
C
2


Page 3
[10] 1.
Multiple Choice
— In each part below, select the
best
single choice, and write your
answer in your test booklet.
No explanations are required; part-marks may be awarded
for partially correct answers.
All parts (i)–(v) below have equal weight.
ψ
(
x
)
x
(position)
ψ
a
:
0
0
L
2
L
2
L
ψ
(
x
)
x
(position)
ψ
b
:
0
0
L
2
L
2
L
i)
Which single statement below is most true of the spatial wavefunctions
ψ
(x)
shown
above?
A.
the expectation kinetic energy of
ψ
a
is greater than that of
ψ
b
B.
the expectation kinetic energy of
ψ
a
is less than that of
ψ
b
C.
the average probability density of
ψ
a
is greater than that of
ψ
b
D.
the average probability density of
ψ
a
is less than that of
ψ
b
E.
exactly two of the above
F.
none of the above
ii)
For the plane wave
ψ
(
x,t)
= exp{
i(kx
ω
t)
} = cos(
kx
ω
t
) +
i
sin(
kx
ω
t
), which single
statement below best describes its probability density distribution?
A.
it is constant and real-valued in space
B.
it is constant and imaginary-valued in space
C.
it is the absolute value |
ψ
(
x)
| of the spatial wavefunction
D.
it is the square modulus |
ψ
(
x)
|
2
of the spatial wavefunction
E.
both A and D
F.
both B and C
iii) For which class of wavefunction below is momentum
p
a well-defined observable
with uncertainty
Δ
p
= 0?
A.
stationary-state wavefunctions of the infinite-potential well
B.
stationary-state wavefunctions of the finite-potential well
C.
plane-wave wavefunctions in free space (–
<
x
<
)
D. simple-harmonic-oscillator wavefunctions
E.
more than one of the above
F.
none of the above


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