# User manual | The Handbook of Mathematics, Physics and Astronomy Data is provided

```The Handbook of Mathematics, Physics and
Astronomy Data is provided
KEELE UNIVERSITY
EXAMINATIONS, 2011/12
Level II
Thursday 12th January 2012, 09.30 – 11:30
PHYSICS/ASTROPHYSICS
PHY-20006
QUANTUM MECHANICS
Candidates should attempt ALL of PART A
and TWO questions from PART B.
PART A yields 40% of the marks, PART B yields 60%.
A sheet of useful formulae can be found on page 8.
NOT TO BE REMOVED FROM THE EXAMINATION HALL
PHY-20006
Page 1 of 8
PART A
A1
Give a physical interpretation of the wave function Ψ in terms of
the observed position of the particle and explain how this leads to
the concept of a normalised wavefunction.
[4]
A2
Explain the following concepts and give an example of a physical
system that demonstrates each concept.
• Wave-particle duality.
• Quantisation of energy.
[4]
A3
Calculate the expectation value for the position, hxi, of a particle
with the normalized wave function
√
Ψ(x, t) = 2π e−πx e−iωt
x > 0.
Z ∞
rk
0
exp(−αr) dr =
k!
αk+1
[4]
A4
State three differences between the predictions of classical physics
and the predictions of quantum mechanics for the properties of a
[4]
particle in a harmonic oscillator potential, V (x) = 12 kx2 .
A5
Sketch the energy eigenfunction, ψ1 and ψ2 , for the ground state
and first excited state of the finite square-well potential






V (x) = 
PHY-20006




0
−V0
0
x < −a
−a ≤ x ≤ a
x>a
Page 2 of 8
[4]
A6
Calculate the expectation value hEi and uncertainty ∆E for the
energy of a particle with the wavefunction
1
1
Ψ(x, t) = √ ψ1 (x)e−iE1 t/h̄ + √ ψ2 (x)e−iE2 t/h̄ ,
2
2
where E1 = 1 eV and E2 = 2 eV.
[4]
A7
Estimate the natural line width in Hz for a transition from an energy
level with a lifetime δt = 10−8 s.
[4]
A8
Explain why identical quantum particles in the same region of space
are also indistinguishable.
[4]
A9
List all possible values of J, the magnitude of the total angular momentum, for an electron with orbital angular momentum quantum
[4]
A10
Explain the origin of “fine-structure” in the spectrum of the hydrogen atom.
[4]
/Cont’d
PHY-20006
Page 3 of 8
PART B
B1
Answer TWO out of FOUR questions
A particle of mass m is trapped in the following potential:






V (x) = 




∞
0
∞
x<0
0≤x≤a
x>a
(a) Show that the solutions of the time independent Schrödinger
equation are



0


 q
ψn (x) = 




2
a
sin(nπx/a) n = 1, 2, 3, . . . ,
0
x<0
0≤x≤a
x>a
and so derive an expression for the energy of the particle in
terms of a and m.
[20]
(b) What is the expectation value for x in this case? Justify your
[3]
(c) Write down the wavefunction, Ψ(x, t), for this particle in terms
of a and m
[3]
(d) Discuss briefly whether the following two statements are consistent with each other.
• The momentum of a particle with kinetic energy E and mass
m is given by p2 = 2mE.
• The expectation value of the momentum for the particle with
the eigenfunction ψn (x) is hpi = 0.
[4]
/Cont’d
PHY-20006
Page 4 of 8
B2
Consider a particle with the wave function Ψ(x, t) = ψ(x)e−iωt where






ψ(x) = 




0
A [1 + cos(πx)]
0
x ≤ −1
−1 < x < 1
x≥1
(a) Normalize this wavefunction.
[8]
(b) Show that the ground state has definite parity and state its
value.
[4]
(c) Calculate the uncertainty in the observed position, ∆x.
[8]
(d) Show that ψ has the required mathematical properties for a
valid wave function at the boundaries x = ±1.
[5]
(e) Discuss whether Ψ(x, t) can be a valid wave function for a particle in a harmonic oscillator potential if it is not one of the
energy eigenfunctions, ψn (x).
[5]
You may use the following standards integrals without proof in your
Z
[1 + cos(x)]2 dx =
Z 1
−1
1
[6x + 8 sin(x) + sin(2x)] + C
4
x2 [1 + cos(πx)]2 dx = 1 −
15
2π 2
/Cont’d
PHY-20006
Page 5 of 8
B3
Consider a particle with mass m and energy E incident on a barrier
with height VB and width a, such that E < VB .
VB
E
0
a
x
(a) Show that the energy eigenfunction
ψ1 = AI eikx + AR e−ikx ,
is a solution of the time-independent Schrödinger equation for
the region x < 0 and hence derive an expression for k in terms
of E.
[10]
(b) What is the physical interpretation of the quantity R =
|AR |2
|AI |2
?
[3]
(c) State an expression for the energy eigenfunction in the region
x > a in terms of an amplitude AT . Explain your answer. [6]
(d) Give one example of a physical process that demonstrates or
2
T|
exploits the fact that T = |A
2
|AI | > 0. State the physical origin
and approximate value of the barrier potential, VB , for your
example.
[6]
(e) Discuss what happens to the value of T in the case that the
mass m is very large.
[5]
/Cont’d
PHY-20006
Page 6 of 8
B4
The wave functions for an electron in a simple model of the hydrogen
atom have the form
Ψn,ℓ,mℓ (r, θ, φ, t) =
u(r)
Yℓ,mℓ (θ, φ)e−iEt/h̄ .
r
The radial eigenfunction for an electron in a hydrogen atom in the
2p state is
!
1
r 2 −r/2a0
u(r) = √
,
e
24a0 a0
where a0 is the Bohr radius.
(a) State the physical quantity most closely associated with each of
the quantum numbers n, ℓ and mℓ , and state the possible values
for each quantum number for an electron in a 2p state.
[6]
(b) Calculate the expectation value, hri, for an electron in the 2p
state. You may use the following standard integral without
Z ∞
rk
0
exp(−αr) dr =
k!
αk+1
[8]
(c) With the aid of a sketch, explain why hri is different from the
most probable observed value of r for the electron.
[4]
(d) With the aid of a labelled diagram, describe the main features
of the Stern-Gerlach experiment. Explain how this experiment
shows that the eigenfunctions Ψn,ℓ,mℓ do not give a complete
description for the properties of an electron in a hydrogen atom.
[12]
/Cont’d
PHY-20006
Page 7 of 8
Quantum Mechanics formulae
Time independent Schrödinger equation
d2 ψ 2m
+ 2 [E − V (x)] ψ = 0
dx2
h̄
PHY-20006
Page 8 of 8
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