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UNSW Sydney
TERM 1 2020 EXAMINATIONS
PHYS2111: Quantum Physics


1. TIME ALLOWED – 2 hours
2. READING TIME – 10 minutes
3. THIS EXAMINATION PAPER HAS 7 PAGES
4. TOTAL NUMBER OF QUESTIONS – 4
5. TOTAL MARKS AVAILABLE – 100
6. MARKS AVAILABLE FOR EACH QUESTION ARE SHOWN IN THE EXAMINATION PAPER
7. ALL ANSWERS MUST BE WRITTEN IN INK. EXCEPT WHERE THEY ARE EXPRESSLY REQUIRED, PENCILS MAY BE
USED ONLY FOR DRAWING, SKETCHING OR GRAPHICAL WORK
8. THIS PAPER MAY BE RETAINED BY CANDIDATE
9. CANDIDATES MAY BRING TO THE EXAMINATION: UNSW APPROVED CALCULATOR
10. THE FOLLOWING MATERIALS WILL BE PROVIDED: Exam booklets
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The following information may be useful
Planck’s constant h = 6.626 × 10−34 Js
Fundamental charge unit e = 1.60 × 10−19 C
Speed of light (vacuum) c = 3.0 × 108 m/s
Electron mass = 9.1 × 10−31 kg = 0.511 MeV/c2
Neutron mass = 1.675 × 10−27 kg = 939.6 MeV/c2
Proton mass = 1.672 × 10−27 kg = 938.3 MeV/c2
Boltzmann’s constant k = 1.38 × 10−23 JK−1
Permittivity constant εo = 8.85 × 10−12 Fm−1
Gravitational constant G = 6.67 × 10−11 Nm2/kg2
Angstrom 1 Å = 1.0 × 10−10 m
h/mec = 2.43 × 10−12 m
1 eV = 1.60 × 10−19 J
1 J = 6.24 × 1018 eV
Time-independent Schrödinger Equation: ( ) ( ) ( )xExV
dx
xd
m
ψψψ =+− 2
22
2

Time-dependent Schrödinger Equation: ( ) ( ) ( )
t
txitxxV
x
tx
m ∂
∂
=+
∂
∂
−
,ψ,ψ)(,ψ
2 2
22


Bohr-Sommerfeld equation: ∮ = ℎ
( ) ( )
b
bxxdxbx
4
2sin
2
sin 2 −=∫
( ) ( )
8
2cos
4
)2sin(
4
sin
2
2 xxxxdxxx −−=∫
( ) ( ) ( )23
23
22
4
2cos2sin
8
1
46
sin
b
bxxbx
bb
xxdxbxx −





−−=∫
b
π2b =∫
∞
∞−
− dxe x
1
0
b !
+
∞
− =∫ nxn b
ndxex
�
2
2
=

�


− �










�( − )1 2⁄ = − 23 ( − )3 2⁄
3


() =

+


( ) cosθsinθ22θsin =
Pauli spin matrices: ≡ �
0 11 0�, ≡ �0 − 0 �, ≡ �1 00 −1�
() = 1
√2ℏ� ()−/ℏ∞−∞
� −
22
∞
−∞
= √


� −
2

∞
−∞
= √
� −(2++)∞
−∞
= �

exp�2 − 44 � , > 0
� 2−
22
∞
−∞
= √23
� 2|Φ()|2∞
−∞
= 2ℏ22
Ground state of the harmonic oscillator: 0() = 1
�0
2�
1
4�
−
2 20
2⁄ , where 02 ≡
ℏ


[A,B] = AB – BA
[A,B]† = [B†,A†]
Table of spin operator actions
σx|uu〉 = |du〉 σx|ud〉 = |dd〉 σx|du〉 = |uu〉 σx|dd〉 = |ud〉
σy|uu〉 = i|du〉 σy|ud〉 = i|dd〉 σy|du〉 = −i|uu〉 σy|dd〉 = −i|ud〉
σz|uu〉 = |uu〉 σz|ud〉 = |ud〉 σz|du〉 = −|du〉 σz|dd〉 = −|dd〉
Note that σ acts on the left letter in the ket. The table for τ is similar but τ acts on the right
letter in the ket instead.
Ladder operators
± = 1
√2ℏ
(∓ + )
+ = √ + 1+1
− = √−1

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Question 1 (25 Marks)
(a) One of the key mathematical concepts in quantum mechanics is that of eigenvectors
and eigenvalues.
(i) In one or two sentences, explain what the terms ‘eigenvector’ and ‘eigenvalue’
mean for some arbitrary operator ̂.
(ii) A fellow student makes the statement “Eigenvector, eigenfunction, eigenstate,
they’re really just different ways of representing the same essential thing.”
Would you agree or disagree? Briefly explain your reasoning (1-2 sentences).
(b) In quantum mechanics, observables are represented by Hermitian operators.
(i) What does it mean for an operator to be Hermitian? In a Hermitian matrix,
what do we automatically know about terms on the diagonal and pairs of off-
diagonal terms symmetric about the diagonal?
(ii) Why do we require observables to be represented by Hermitian operators in
quantum mechanics?
(c) In the course we have often considered a ‘quantum spin’, the quantum equivalent of a
particle with angular momentum. You learn about classical angular momentum in first
year. Briefly discuss how the outcome of measuring the z-component of the angular
momentum differs between the classical and quantum cases. Marks will be allocated
based on the conciseness and effectiveness of your answer rather than length alone.
Stick to what is important in answering this question.
(d) For a quantum spin system with spin ½, all possible spin states can be represented in a
two-dimensional complex vector space with basis vectors |⟩ = �10� and |⟩ = �01�.
(i) Using the Pauli matrix � given in the equation sheet, explain briefly what the
two possible outcomes for measuring the z-component of the spin would be. In
your answer, make it clear what the two eigenvalues and their corresponding
eigenvectors mean physically.
(ii) Suppose a quantum spin is in the state |⟩ = 1
√6
�1+2 � . If you make an
observation of the z-component of the spin �, then what are the respective
probabilities the two possible outcomes? Show that these add to 1 as expected.
(iii) For the x-component of the spin � , find the eigenvectors |⟩ and |⟩ that
correspond to the two possible measurement outcomes. Be sure your final
eigenvectors are properly normalized. (n.b., there is more than one way to do
this; one is faster than the other).
(iv) The vectors you obtain in (iii) are not |⟩ and |⟩. Is this a problem? Explain
your answer in no more than 2-3 sentences.
(v) Calculate the respective probabilities of measuring the two observable
outcomes for the x-component of the spin for |⟩. Show these add to 1.
(vi) Calculate the expectation value for σx, which is obtained as 〈〉 = ⟨||⟩.
Show you can get the same result using the two probabilities obtained in (iv).

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Question 2 (25 Marks) (a) A commutator of two operators ̂ and � is defined as �̂,�� = ̂� − �̂.
(i) What does it mean mathematically for two operators to ‘not commute’? In 1-2
sentences, explain the physical implications for non-commuting observables.
(ii) Show that the commutator �̂,�� is Hermitian if ̂ and � are both Hermitian.
(iii) Show that [σx,σy] = 2iσz.
(b) For two observables L and M we can obtain the generalized uncertainty principle as:
ΔΔ ≥ �
2 〈�� ,��〉�
(i) Assume a single spin prepared in the state |⟩. Use the generalized uncertainty
principle to show that ∆σx∆σy ≥ 1.
(ii) Suppose instead that the single spin is in the state |⟩ = |⟩
√2
+ |⟩
√2
. What will the
uncertainty relation ∆σx∆σy become? Briefly explain the physical meaning.
Note: For each of (i) and (ii) above you will need 〈σz〉, it is acceptable to briefly explain
the value you use for this rather than calculate it explicitly.
(c) Assume Charlie has prepared two spins in the singlet state |Ψ⟩ = 1√2 (|⟩ − |⟩).
(i) Alice now measures σz and Bob measures τz. What is the expectation value of
σzτz? The table of spin operator actions in the equation sheet will be helpful here.
Note that Alice’s operator σ acts on the left letter in the ket and Bob’s operator τ
acts on the right letter in the ket. The table for τ is not given but should be easy
to infer.
(ii) What does your result in (i) say about the outcomes of Alice and Bob’s
measurements of their spins if they measure their respective z-components?
(iii) There isn’t enough time to make you calculate this, but 〈σxτy〉 = 0 and not −1
(you can confirm it for yourself after the exam if you don’t believe me). What
does this mean when you consider this means Alice measuring the x-component
of her spin and Bob measuring the y-component of his spin.
(d) There is an interesting postulate in quantum mechanics that says any 2 × 2 Hermitian
matrix L can we written as the sum of the three Pauli matrices and the identity matrix:
� = + + +
where a, b, c and d are all real numbers. Verify this postulate.

Question 3 (25 marks)
(i) Consider the Hamiltonian for the harmonic oscillator
H =
1
2m
[
p2 + (mωx)2
]
, (0.1)
and the ladder operators
a± =
1√
2 ~mω
(∓i p+mω x) , (0.2)
where p ≡ −i~ ∂
∂x
is the momentum operator. Verify the following equations:
(a) [a−, a+] = 1.
(b) H = ~ω
(
a+a− + 12
)
.
(ii) Knowing that a−ψ0(x) = 0 for the ground state, derive ψ0(x) (don’t forget to
normalize it).
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Question 4 (25 marks)
(i) Consider the stationary states ψn(x), with n = 1, 2, 3, ..., for the infinite square
well potential and the corresponding energy levels En =
n2pi2~2
2ma2
.
• Knowing that the ψn(x) are orthonormal, find Ψ(x, t) for the following initial
conditions:
Ψ(x, 0) = A (4ψ1 + ψ2) , (0.3)
where ψ1 is the ground state, ψ2 is the first excited state and A is a constant.
For this question, you DO NOT need to specify what the explicit form of the ψn
functions is but you DO NEED to determine A.
• What is the probability that a measurement of the energy returns the value
E = 9pi
2~2
2ma2
? Explain your answer.
(ii) Consider the following potential
V (x) =
{
0 if x < 0 ,
−V0 if x > 0
(0.4)
where V0 > 0, and a particle of mass m and energy E > 0 approaching it from the left
(i.e. from x < 0). What is the value of the reflection coefficient for E = 2V0?
END OF EXAM
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