The Handbook of Mathematics, Physics and Astronomy Data is provided

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
Answer all TEN questions
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.
You may use the following integral without proof in your answer.
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
number ℓ = 2. Give your answers in units of h̄.
[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
answer. (Hint: No calculation required.)
[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
answers.
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
proof in your answer.
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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