2012_Test_1_quantum.pdfFamily Name (Please print ) Givenshowing page 13 out of 5
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 (nonprogrammable, nongraphic 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
=
hν
=
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
)Ψ
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
mω
2
x
2
s
=
mω
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; partmarks 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 realvalued in space
B.
it is constant and imaginaryvalued 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 welldefined observable
with uncertainty
Δ
p
= 0?
A.
stationarystate wavefunctions of the infinitepotential well
B.
stationarystate wavefunctions of the finitepotential well
C.
planewave wavefunctions in free space (–
∞
<
x
<
∞
)
D. simpleharmonicoscillator wavefunctions
E.
more than one of the above
F.
none of the above