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UNIVERSITY OF SOUTHAMPTON

PHYS6003W1

SEMESTER 1 EXAMINATION 2014-2015 ADVANCED QUANTUM PHYSICS Duration: 120 MINS (2 hours)

This paper contains 9 questions.

Answer all questions in Section A and only two questions in Section B. Section A carries 1/3 of the total marks for the exam paper and you should aim to spend about 40 mins on it. Section B carries 2/3 of the total marks for the exam paper and you should aim to spend about 80 mins on it. An outline marking scheme is shown in brackets to the right of each question. A Sheet of Physical Constants is provided with this examination paper. Only university approved calculators may be used. A foreign language translation dictionary (paper version) is permitted provided it contains no notes, additions or annotations.

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Section A A1. Write down the adjoint of the following expression involving bras and kets, in

ˆ and Λ ˆare operators: which ai are complex scalars, and Ω ˆ + a4 ˆ Λ ˆΛ a1∗ |vi + a2 |wih p|qi + a3 Ω |ri |ui . A2. Consider the arbitrary ket |ui =

Pn

i=1 ui |ii,

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where {|ii} is an orthonormal basis.

Show that ui = hi|ui for all values of i. Using this result, then prove that

Pn

i=1 |iihi|

= Iˆ, where Iˆ is the identity operator.

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A3. The kets in the set {|ii} have non-zero (and finite) norm, and are mutually

orthogonal (i.e. hi| ji = 0 for i , j). Show that they form a set of linearly

independent vectors.

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A4. The φ dependent part of the position space wavefunction (in spherical coordinates) of an eigenvector of orbital angular momentum is given by eimφ , where m is the quantum number associated with the z component of orbital [3]

angular momentum. Show that m has to be an integer. A5. Explain the difference between a classical bit and a qubit, giving an example of

[3]

the latter.

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Section B B1. A simple harmonic oscillator in one dimension is defined by the Hamiltonian

Hˆ = h¯ ω(a† a + 1/2) , where the raising and lowering operators are given respectively by

a† =

r

a=

r

mω i pˆ , xˆ − √ 2¯h 2mωh¯

i mω xˆ + √ pˆ , 2h¯ 2mωh¯

with [ xˆ , pˆ ] = ih¯ and [a, a† ] = 1. Let |ni be a normalised (i.e. hn|ni = 1)

ˆ eigenvector of the Hamiltonian with an energy eigenvalue of En (i.e. H|ni = En |ni). ˆ a] = −h¯ ωa and [H, ˆ a† ] = h¯ ωa† . (a) Show that [H,

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(b) Using the commutation relations in (a), show that a|ni is an eigenvector of Hˆ with eigenvalue En − h¯ω, and that a† |ni is an eigenvector of Hˆ with

eigenvalue En + h¯ ω.

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(c) Using the relations

a† |ni =

√ √ n + 1|n + 1i and a|ni = n|n − 1i,

calculate explicitly the expectation values h pˆ i, h pˆ 2 i, h xˆ i and h xˆ 2 i for a

simple harmonic oscillator in the state |ni. Obtain an expression for the

product ∆x∆p as a function of n (where ∆x =

p h xˆ2 i − h xˆi2 etc), and

confirm that it complies with Heisenberg’s uncertainty relation.

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(d) Now consider the eigenvectors |ni and |mi where n , m. Find the values

of n and m for which the inner product hn| xˆ 2 |mi vanishes and the values for

which this inner product does not vanish (show your working).

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B2. Consider the following relations for angular momentum operators:

Lˆ − = Lˆ x − i Lˆ y ,

Lˆ + = Lˆ x + i Lˆ y Lˆ +|lmi = h¯

p (l − m)(l + m + 1)|l, m+1i Lˆ z |lmi = mh¯ |lmi

Lˆ −|lmi = h¯

p (l + m)(l − m + 1)|l, m−1i ,

Lˆ 2 |lmi = l(l + 1)h¯ 2 |lmi .

For a system with orbital angular momentum l = 1 the eigenvectors |lmi can be labelled by the eigenvalue m alone, and so one has three eigenvectors |1iz ,

|0iz , and | − 1iz for m = 1, 0, −1 respectively.

(a) Using the above relations for the angular momentum operators, show that the matrix representation of Lˆ x for l = 1 in the basis of eigenvectors of Lˆ z and Lˆ 2 is given by



0 1 0



h¯   Lˆ x = √  1 0 1  . 2 0 1 0

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(b) Now find the eigenvalues and (normalised) column vector representations of the eigenvectors of Lˆ x , neglecting global phase factors.

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(c) Calculate the probability that a measurement of Lˆ x will give zero for a system that is in the state



1



1   |ψi = √  2  . 14 3

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(a) State the four postulates of quantum mechanics.

For concreteness,

consider a one-dimensional system e.g. a particle moving along the x[6]

axis.

ˆ in an n dimensional Hilbert space has eigenvectors |ωi i (b) An operator Ω

with corresponding eigenvalues ωi , where i is an integer from 1 to n. The

ˆ is denoted by hΩi ˆ . Starting from the expression expectation value of Ω ˆ = hΩi

n X

P(ωi ) ωi ,

i=1

where P(ωi ) is the probability of obtaining the eigenvalue ωi from a

ˆ , show that hΩi ˆ can be written as measurement of Ω ˆ = hψ|Ω|ψi ˆ , hΩi [5]

where |ψi is the state vector for the system. (c) Explain what is meant by an “eigenbasis” of a Hilbert space. Now consider

ˆ in (b) above being the case of two of the eigenvalues of the operator Ω degenerate i.e. ω1 = ω2 = ω. Show that there are an infinite number of [4]

eigenbases for this Hilbert space.

ˆwith eigenvectors |λi i, and correspond(d) Now consider another operator Λ

ˆcommutes with ˆ Ω ing eigenvalues λi with no degeneracy. Suppose that Λ

,ˆˆ Ω] = 0. Show that {|λi i} are also eigenvectors of Ωˆ . i.e. [Λ

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(a) Write down an expression for the most general state (|ψiAB) in a Hilbert space VA ⊗ VB, where VA and VB are Hilbert spaces with orthonormal

bases {|iiA } and {| jiB} respectively. Then give the condition for this general

state to be separable and the condition for this state to be entangled.

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(b) Suppose that Alice has a single qubit state |φi = α|0i + β|1i which she

wishes to teleport to her friend Bob. To do this she creates an entangled state

1 |ψi = √ (|00i + |11i) , 2 keeping the first (left) qubit and sending the second (right) qubit to Bob. Show that twice the product state |φi|ψi can be written as

2|φi|ψi = |B0 i(α|0i + β|1i) + |B1 i(α|1i + β|0i) +|B2 i(α|0i − β|1i) + |B3 i(α|1i − β|0i), where

1 |B0 i = √ (|00i + |11i) , 2

1 |B1 i = √ (|01i + |10i) , 2

1 |B2 i = √ (|00i − |11i) , 2

1 |B3 i = √ (|01i − |10i) . 2

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(c) Now show that the operators

Iˆ = |0ih0| + |1ih1| , Yˆ = |0ih1| − |1ih0| ,

Xˆ = |0ih1| + |1ih0| , Zˆ = |0ih0| − |1ih1| ,

can be used to transform the single qubit parts of the product state above into |φi. That is, show that

ˆ α|0i + β|1i) = |φi , I(

ˆ X(α|1 i + β|0i) = |φi ,

ˆ Y(α|1 i − β|0i) = |φi ,

ˆ Z(α|0 i − β|1i) = |φi .

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(d) Explain step by step how Alice and Bob can make use of the above results to devise a scheme for teleporting the original qubit |φi without either of them knowing its state.

END OF PAPER

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