Write your CANDIDATE NUMBER clearly on each of the THREE answer books provided. Hand in THREE answer books even if they have not all been used.


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1 UNIVERSITY OF LONDON BSc/MSci EXAMINATION June 2007 for Internal Students of Imperial College of Science, Technology and Medicine This paper is also taken for the relevant Examination for the Associateship QUANTUM MECHANICS For SecondYear Physics Students Monday 4th June 2007: to Answer THREE questions. All questions carry equal marks. Marks shown on this paper are indicative of those the Examiners anticipate assigning. General Instructions Write your CANDIDATE NUMBER clearly on each of the THREE answer books provided. If an electronic calculator is used, write its serial number in the box at the top right hand corner of the front cover of each answer book. USE ONE ANSWER BOOK FOR EACH QUESTION. Enter the number of each question attempted in the horizontal box on the front cover of its corresponding answer book. Hand in THREE answer books even if they have not all been used. You are reminded that the Examiners attach great importance to legibility, accuracy and clarity of expression. University of London / 2 / 210 Turn over for questions
2 1. A particle of mass m is confined by a potential V(xsuch that V(x = x >a V(x = 0 a x a. Show that the normalised energy eigenfunctions for this system are u n = 1 a sin nπx 2a (for n = 2, 4, 6... and u n = 1 a cos nπx 2a (for n = 1, 3, [9 marks] (ii If the particle is in its ground state evaluate (by appropriate integration the following: (a The expectation value of x. [2 marks] (b The expectation value of the momentum p. Note: The momentum operator is given by ˆp = i h d dx. (c The expectation value of p 2. [2 marks] Without performing any integration use the Heisenberg uncertainty principle and the answers to part (ii above to estimate the expectation value of x 2 (you may make the approximation that the ground state for this potential is a minimum uncertainty state
3 2. A particle of mass m in a harmonic potential given by V 1 (x = 1 2 mω2 1 x2 is not in an energy eigenstate but rather is described by the normalised wavefunction ( 1/4 2mω1 ψ = e mω 1x 2 / h. π h By writing down the appropriate overlap integral show that the probability that a measurement of the particle s energy will give the result E = 3 hω 1 /2 is zero. Note: The lowest normalised energy eigenstates for the general harmonic oscillator potential V(x= 1 2 mω2 x 2 are u 0 = (mω/π h 1/4 exp( mωx 2 /2 h u 1 = (4/π 1/4 (mω/ h 3/4 x exp( mωx 2 /2 h u 2 = (mω/4π h 1/4 [2(mω/ hx 2 1] exp( mωx 2 /2 h. (ii Evaluate the probability P 1 that a measurement of the particle s energy will give the result E = 5 hω 1 /2. [6 marks] The potential is suddenly made steeper so that it is described by the potential V 2 (x = 1 2 mω2 2 x2 with ω 2 >ω 1. We define P 2 to be the probability that a measurement of the particle s energy made after this sudden change will yield the ground state energy of the new potential. Show that P 2 = 2(ω1 ω 2 1/2 ω 1 + ω 2 /2. [5 marks] (iv Under what circumstances is P 2 = 1? In this case what kind of function is ψ with respect to V 2 (x? [5 marks] You may require the standard integrals exp( ax 2 dx = a, x 2 exp( ax 2 dx = 1 2a a Please turn over
4 3. A particle is described by the wave function ψ(x = Ae x2 /w 2. Normalise this wave function. You may require the standard integral: exp( ax 2 dx = a. (ii Sketch ψ 2. At what values of x is ψ 2 equal to e 1/2 times its maximum value? If the momentum p of the particle is measured the probability of finding p in the range dp is P(pdp. Show that P(p = 2 w π 2 h e p2 w 2 /2 h 2. You may require the standard integral e x(ax+ib dx = a e b2 /4a. (iv Sketch P(pand determine what values of p correspond a probability density which is e 1/2 times its maximum value. (v How do these results relate to the Heisenberg uncertainty principle? (vi A laser is adjusted to give a circular output beam travelling along the z axis whose angular divergence is as small as possible. The intensity profile in the x direction of the emerging beam is given by I(x = Be 2x2 /w 2. Since a laser beam is comprised of photons you may assume that I(x ψ(x 2 where ψ(x is defined above. If λ = 2π/k = 405nm and w = 1mm estimate the size of the spot formed on a wall at a distance of 10 km from the laser output port
5 4. In cartesian coordinates the operator ˆL 2 = ˆL ˆL, where ˆL is the quantum mechanical operator for angular momentum, can be written as ˆL 2 = ˆL 2 x + ˆL 2 y + ˆL 2 z, where the components of ˆL obey the commutators [ ˆL x, ˆL y ]=i h ˆL z, [ ˆL y, ˆL z ]=i h ˆL x, [ ˆL z, ˆL x ]=i h ˆL y. Show that You may assume without proof that [ ˆL 2 x, ˆL z ]= i h( ˆL x ˆL y + ˆL y ˆL x. [ ˆL 2 y, ˆL z ]=i h( ˆL x ˆL y + ˆL y ˆL x. Finally by considering [ ˆL 2 z, ˆL z ] show that [ ˆL 2, ˆL z ] = 0. By similar reasoning it is possible to show that [ ˆL 2, ˆL x ]=[ˆL 2, ˆL y ]=0 (do not prove this. What general conclusions about angular momentum can you draw from these commutation relations? [7 marks] (ii Raising and lowering operators for angular momentum can be defined as Show that ˆL ± = ˆL x ± i ˆL y. [ ˆL z, ˆL ]= h ˆL. (iv If φ m is an eigenfunction of ˆL z with eigenvalue m h show that ( ˆL φ m is also an eigenfunction of ˆL z but with eigenvalue (m 1 h. In cartesian coordinates ˆL x =ŷ ˆp z ẑ ˆp y, ˆL y =ẑ ˆp x ˆx ˆp z, ˆL z =ˆx ˆp y ŷ ˆp x. Show that φ 1 = (x iy 2 and φ 0 = z, (v are eigenfunctions of L z with eigenvalues h and 0 respectively. Verify that the effect of ˆL on φ 0 is as predicted in part above Please turn over
6 5. The Coulomb potential for an electron bound to a proton is given by e2 V C = 4πɛ 0 r. An electron in such a potential is described by the normalised wave function ψ(r = R(rƔ(θ, φ. Where R(r and Ɣ(θ, φ are themselves normalised functions given by R(r = ( 1 24a 3 0 1/2 r a 0 e r/2a 0, Ɣ(θ, φ = 1 4π [ cos θ sin θe iφ ]. Express Ɣ(θ, φ as a superposition of angular momentum eigenstates. (ii What are the possible results of measurements of L 2 and L z? What is the probability that a measurement of L z will yield the result h? [1 mark] (iv Evaluate the expectation value of L z. [2 marks] (v (vi Calculate the expectation value of the electronproton separation. Calculate the expectation value of the potential energy of the electron. [5 marks] [5 marks] You may require the following information: Normalised spherical harmonics 3 3 Y 10 = 4π cos θ, Y 1±1 = 8π sin θe±iφ. Standard Integral 0 r n e r/a dr = n!a n+1. Bohr radius a 0 = m
7 6. The energy of a free electron in an externally applied magnetic field depends upon its spin state. If the field is along the xaxis the relevant Hamiltonian is given by H = eb m Ŝx and the energy eigenvalue equation is Hχ ± = E ± χ ±, where E ± are energy eigenvalues and χ ± are eigenvectors (eigenspinors. Show that the normalised eigenvectors χ ± can be written as χ + = 1 ( 1, χ 2 1 = 1 ( and find the energy eigenvalues E ±. (ii (a ( i At time t = 0, a particle is in the state χ = 0 (b Express χ as a superposition of χ ±. [8 marks]. Show that χ is normalised. [1 mark] (c What is the expectation value of Ŝ z at time t = 0? [2 marks] The state χ given above is not a stationary state but rather it evolves periodically in time. Write down an expression for χ(t and hence an expression for Ŝ z (t, the time dependent expectation value of Ŝ z. Show that Ŝ z (t first equals zero at time t = πm/2eb. [6 marks] Notes: Energy eigenstates χ n with eigenvalues E n evolve in time as χ n e ient/ h. The operators Ŝ x, Ŝ y and Ŝ z are given by Ŝ x = h ( , Ŝ y = h ( 0 i 2 i 0, Ŝ z = h ( End 40811
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