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XPHC 2663 / XPHA 2663

SECOND PUBLIC EXAMINATION

Honour School ofPhysics Part A: 3 and 4 Year Courses

HonourSchool ofPhysicsand Philosophy Part A

A3: Quantum Physics

Friday, 20 June 2003, 9.30 am — 12.3Opm

Answerall ofSection A andthree questionsfromSection B.

Start the answer to each question on afresh page.

A list ofphysical constants and conversionfactors accompaniesthispaper.

The numbers in the margin indicate the weightwhich the Examiners expectto

assign to each partof the question.

Do NOTturn over until told that you may do so.

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Section A

1. Explain which aspects of the photo-electrzc effect cannot be accounted for byclassical

physicsand how

Quantum Theoryresolvesthe difficulties. [7]

2. Showthat the matrices

h(0 i\ h(0 —'\ Ii(i o\Sx=i o)' 2\i a)' —i)'

satis& the commutation rules for angular momentum. Find the eigenvalues andnormalizedeigenvectorsof 8. [8]

3. A non-relativistic free particle of mass in moving in one dimension haswavefunction

%b(x,t) = Aexp (k (px—

Etp)t))+ARexp(_ px+

where A and R are constants. Find E(p). At time t= 0 themomentum of he particleis measured. What are the possibleoutcomes and their respective probabilities? [6]

4. An electron is in a state withorbital angular

momentumquantum

number £.What valuesmay its totalangular momentumquantumnumberj take when

(a) £ = 0,

(b) 1=1? [4]

5. Describe the Zeeman Effect. What magnetic fluxdensityis required to producea Zeemansplittingof0 05cm1 in theground stateof hydrogen? [6]

6. Determinewhether hefollowing matrices represent arotation in threedimensions

and, ifso, find the angle and axis of rotation:

1 1

11 '?'7 1

(a) — 1 —"/ I; (b) — _L 1_L. —162 0)2

[9]

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Section B

7. Two non-interacting particles of the same mass m, and coordinates x1 and x2

respectively, occupy the same one-dimensional potential wellV(x).

Theenergyeigenfunctionsfor a single particle in the wellare denoted by çb(x), withn = 1,2,

and the corresponding energies, which are non-degenerate, by E. Write down theHamiltonian for the two-particle system and show that

d1fl1,fl2(Xi,X2)= fl1(x1)fl2(x2)

are energy eigenfunctions. What is the corresponding energy eigenvalue? [4]

Explain briefly the meaningof exchange symmetry. [3]

In the following cases, state the degeneracyof the ground state and of the first

excited state, and express thetime-independent wavefunction for eachof hese states interms ofthe functions x2):

(a) the particles are not identical and have spinzero;

(b) the particles are identical and have spinzero [6]

Assume nowthattheparticles are identicalandhavespin What arethepossiblespin states for the two particle system? What is the degeneracy of the ground state,and ofthe firstexcited state? Express the time-independent wavefunctionsfor all thesestatesin termsofthe functions fl1 fl2(xi,x2) and he spinstates. [7]

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8. Explain why aparticlewhich is in an energy eigenstate cannot be moving in the

classical sense. [3]

The simple harmonic oscillator has Hamiltonian H with eigenstates I) and

corresponding eigenvalues E = hw(n+ ), where n = 0, 1 At t = 0 theparticle is in the state

JW(t=0)) =(I0)+I1))Show that at subsequent times the stateofthe particle is

IW(t)) =_= (e_0t/h 0) + e_1t/hi

Ii))

Thetime-independent avefunctionsor the ground and first excited states of the

simple harmonic oscillator are

O(x) =() 2/'2 and &(x) = ()* e_x2/2a2,

where a2 = h/mw Show hat

(W(t)IxI'h'(t)) =

and calculate ('I'(t)IpjW(t)). [10]

Find

mw2(W(t)IxIW(t)),

and comment on yourresult [2]

9. What is the physical origin of the spin-orbit interaction? For the electron in a

hydrogenatom the spin-orbit contribution to the Hamiltonian is

21so=— .s,

4ir2me r

where r is the radial coordinate of the electron and 1 and s are the orbital and spinangular momentum operators respectively. Justify the form of this expression (Youare not required to explain the originofg = 2 for the electron.) [10]

What is meant by the terms conserved quantity and good quantum number? [4]

Consider the electron in a hydrogen atom in the approximation that theHamiltoman is given by

e2 +H2me 4ireor

Is the z-component of orbital angular momentum a conserved quantity? You may

assume that [p2, 1] = 0. What is the implication ofyourresult for the classificationofthe energy levelsofthe hydrogenatom? [6]

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10. What is the purpose ofperturbatzon theory? [2]

A non-degenerate system has Hamiltonian H0 whose eigenfunctions arewithcorresponding eigenvaluesE, wheren=1,2.... The Hamiltonian is modifiedby

the additionofa term H1. Derive an expression for the first-order shift, iE0, in theenergy ofthe ground state. [8]

The one-dimensional infinite square wellpotential

Vo(x) = 0,

Vo(x) = oo, otherwise,

is modifiedby the additionofthe perturbation

Vi(x) = v,

V1(x) = 0, otherwise,

where v is a constant. Show that all the energy levelsof he system are shifted in firstorder by the same amount. [6]

By consideringthe first-order shifted wavefunction given by

(k V1 çbn)'çbçb+ ,'n-'k

show that first-order perturbation theoryis reliable, provided

3h2ir22'2ma

where m is the massof he particle in the well [4]

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