STRUCTURE OF MATTER, VIBRATIONS AND WAVES, AND QUANTUM PHYSICS

Imperial College London BSc/MSci EXAMINATION June 2008 This paper is also taken for the relevant Examination for the Associateship STRUCTURE OF MATTER, VIBRATIONS AND WAVES, AND QUANTUM PHYSICS For 1st-Year Physics Students Wednesday, 11th June 2008: 10:00 to 12:00 Answer ALL questions from Section A, ONE question from Section B, ONE question from Section C and ONE question from Section D. Marks shown on this paper are indicative of those the Examiners anticipate assigning. General Instructions Complete the front cover of each of the 6 answer books provided. If an electronic calculator is used, write its serial number at the top of the front cover of each answer book. USE ONE ANSWER BOOK FOR EACH QUESTION. Enter the number of each question attempted in the box on the front cover of its corresponding answer book. Hand in 6 answer books even if they have not all been used. You are reminded that Examiners attach great importance to legibility, accuracy and clarity of expression. c Imperial College London 2008 P.1.3 1 Turn over for questions

SECTION A 1. (i) The piston of a car engine compresses the mixture of fuel and air, which is initially at 20 C and 1 atmosphere, to 1/10th its initial volume. If the pressure after compression is 23.0 atmospheres, calculate the temperature of the compressed fuel, assuming it is an ideal gas. [2 marks] (ii) The first law in differential form for a gas is, du = dq P dv Use this to show that for an ideal gas, the molar specific heats are related by, C Pm C Vm = R (iii) A microscopic system has only two energy states with potential energies E and E + ΔE. If a total of N such systems are prepared in thermal equilibrium at temperature T, obtain expressions for the population of both levels. [Total 8 marks] P.1.3 2 Please turn over

2. (i) Sketch on the same axes the amplitude of vibration, A(ω), for a forced damped simple harmonic oscillator as a function of driving frequency, ω, for a low and a high Q-factor oscillator. Mark the relative positions of the natural (undamped) frequency, ω 0, and the resonant frequency, ω res. (ii) Waves propagate in a medium where the phase velocity is inversely proportional to wavelength. Show that the group velocity is twice the phase velocity. [2 marks] (iii) The E-string on a violin has a mass per unit length of μ = 4 10 4 kg m 1 and a length of L = 0.33 m. What tension is required to tune it to its correct frequency of f = 660 Hz for its lowest order (n = 1) standing wave mode? [Total 8 marks] P.1.3 3 Please turn over

3. (i) (a) For a photoelectric emission experiment, write down the energy balance equation linking the stopping potential, V 0, the work function, W, and the energy hν of the incident radiation (where h is Planck s constant and ν is the frequency of the radiation). Explain these terms. (b) A stopping potential, of V 0 = 1.1 volts is measured. What is the maximum velocity of the emitted electrons? (ii) (a) Consider the complementary process to the photo-electric effect, namely production of X-rays by electron bombardment of a metallic surface. Write down the equation linking the energy of the incident electrons (expressed in terms of the accelerating voltage, V AC ) and the minimum wavelength of the emitted photons, λ m. (b) If the incident electron energy is 100 ev, what is the minimum wavelength, λ m, of the broad-band (so-called Bremsstrahlung ) X-ray radiation emitted? (c) Briefly describe the difference between the mechanisms that produce this broad-band radiation and the sharper X-ray emission lines that are also observed. (iii) (a) Write down Heisenberg s Uncertainty Principles for momentum-position, and for energy-time. (b) A fundamental particle has a rest energy of 3.097 10 9 ev, and is unstable, with a lifetime of 7.6 10 21 s. Calculate the uncertainty in the rest energy of the particle, as a fraction of the rest energy. [Note: You will need the ev to J conversion, 1.602 10 19 J/eV] [Total 9 marks] P.1.3 4 Please turn over

SECTION B 4. The probability that the x-velocity component of particles in a gas lie in the range v x to (v x and dv x ) is P [v x ]dv x = Ae mv2 x/2k B T dv x ( ) m 1/2. where A = 2πk B T (i) Explain (without proof) how this result follows directly from the Boltzmann relation, which gives the probability of a state of energy E being occupied, given by P [E] e E/kBT. In particular, explain the origins of the exponent and the constant A. [4 marks] (ii) Calculate the mean kinetic energy carried by particles in the x-direction in terms of the temperature T for this system. You may use the following standard integral (which you can quote without proof), + ( π α 3 ) 1/2 x 2 e αx2 dx = 1 2 [6 marks] (iii) Since each velocity component is independent, the probability that a particle has a velocity with components v x, v y, v z is just the product of the independent probabilities, i.e. P [v x, v y, v z ] dv x dv y dv z = P [v x ]dv x P [v y ]dv y P [v z ]dv z Using the expression above, write out a value for P [v x, v y, v z ]. [2 marks] (iv) By rewriting the integral over cartesian coordinates in terms of spherical coordiates, and integrating over the angular coordinates, show that the speed distribution is given by ( ) m 3/2 P [v]dv = 4π v 2 e mv2 /2k B T dv 2πk B T [6 marks] (v) Prove that the average total kinetic energy is 3 2 k BT, and relate this to the value found in part (ii). You may make use of the standard integral, + ( π α 5 ) 1/2 0 x 4 e αx2 dx = 3 8 [7 marks] [Total 25 marks] P.1.3 5 Please turn over

5. (i) Show that the work done on an ideal gas undergoing a quasi-static isothermal compression from V 0 to V 1 (< V 0 ) is given by, W = Nk b T ln(v 0 /V 1 ) [4 marks] (ii) In an external (Stirling) combustion engine, the following processes take place, a) an enclosed gas is compressed by a piston isothermally from volume V 0 to V 1 at temperature T 0. b) the gas is heated by an external source which raises its temperature almost instantaneously (i.e. without volume change) to T 1. c) the gas is allowed to expand again isothermally (i.e. still in contact with the heat source) until its volume returns to V 0. d) the expanded gas is removed from contact with the heat source, and is allowed to cool to temperature T 0 whilst maintaining its volume V 0. Draw this cycle on a PV diagram indicating each process clearly, assuming the enclosed gas is ideal. (iii) Calculate the work done by the gas in this cycle, in terms of N, T 0, T 1, V 0 and V 1. (iv) Calculate the work done over a cycle by a 1.0 litre (= 10 3 m 3 ) engine in which the compression ratio is 10 and the fuel mixture goes from standard temperature and pressure (273 K, 1 atmosphere = 1.01 10 5 Pa) to a temperature of 500 K. (v) Calculate the amount of heat required in taking the gas from temperature T 0 = 273 K to T 1 = 500 K at constant volume. Compare this to the work done in part (iv). Does this imply that the work done is greater than the heat put into the system. If not, explain why? [6 marks] [Total 25 marks] P.1.3 6 Please turn over

SECTION C 6. The equation of motion for liquid in a U-shaped tube can be written as LAρ d2 x dt = 2Aρg x 2 where the real part of x is the displacement of the fluid from equlibrium, L is the total length of liquid in the tube, A is the cross-sectional area of the tube, ρ is the density of the liquid and g = 9.8 m s 2 is the acceleration due to gravity. (i) Show that the equation above has a solution x = x 0 exp [i (ωt + φ)], where x 0 and φ are both real. At what frequency, ω 0, does the liquid oscillate? [4 marks] (ii) Find x 0 and φ given that at t = 0 the liquid is a distance h above the equilibrium position on the right hand side of the tube and initially stationary. [4 marks] (iii) Show that the kinetic energy of the liquid is given by Aρgh 2 sin 2 (ω 0 t). By considering the maximum value of the kinetic energy, obtain an expression for the potential energy of the liquid as a function of time. Liquid moving in the U-tube actually experiences a damping force, with damping constant r = 8πηL, where η is the viscosity of the fluid. (iv) Write down the new equation of motion containing the damping force. By comparing this with the equation of motion for a damped mass on a spring, show that the condition for light damping is ( 4πη Aρ ) 2 < 2g L [6 marks] (v) A U-tube with A = 1.0 10 4 m 2 and L = 1.0 m is filled with ethanol at 20 C which has a density ρ = 790 kg m 3 and a viscosity η = 1.2 10 3 N m 2 s. Show that the oscillations are lightly damped. Assuming the oscillations are very lightly damped, calculate the Q-factor for the system. For what numerical value of A does the oscillation become critically damped? [6 marks] [Total 25 marks] P.1.3 7 Please turn over

7. A wave generator has been designed to make water waves in a swimming pool. It consists of a plate under the water surface driven up and down by a hydraulic piston that is anchored to the bottom of the pool. (i) The vertical displacement of the plate is given by x(t) = A cos (ωt + φ). Obtain expressions for the velocity v(t) and acceleration a(t) of the plate in the vertical direction. Determine A and φ given that at t = 0 the displacement is 0.3 m and the velocity is zero. What is the full range of displacement? (ii) The wave generator creates surface water waves of frequency ω. The phase velocity of such waves is given by v p = gλ 2π where g is the acceleration due to gravity and λ is the wavelength. Show that v p = g/ω. [4 marks] (iii) Two wave generators WG1 and WG2 are installed in a pool spaced by a distance d. Each is operated with the same amplitude A and frequency ω and they oscillate in phase. A swimmer is swimming parallel to them (along the y-axis) a distance L d away. A top view is shown on the next page. Δr is the path difference. P.1.3 8 [This question continues overleaf... ]

Show that the swimmer will experience a maximum wave amplitude at angles given by θ n (max) 2nπv p ωd where n is an integer. What phenomenon is taking place? [6 marks] (iv) Hence calculate the distance the swimmer swims between points of maximum amplitude. Take L = 10 m, d = 1.56 m and ω = 2π rad/s. [6 marks] (v) The swimmer is at point O on the centre line between the wave generators. WG1 is switched off. The swimmer notices that the amplitude of the waves has decreased. By what factor has the intensity of the waves decreased? Explain your answer. [4 marks] [Total 25 marks] P.1.3 9 Please turn over

SECTION D 8. (i) Write down the one-dimensional, time-independent form of the Schrödinger equation, describing the motion of a particle along the x-axis, in the presence of a potential energy U(x). Describe the physical meaning of each term, and of the wave function ψ(x). (ii) (a) In the case of a free particle, for which U(x) = 0 for all x, show that a spatial wave function of the form Ae ikx is a solution to the Schrödinger equation. (b) If the particle is an electron with a mass, m e = 9.1 10 31 kg, moving at a speed of 10 5 m s 1, what is its de Broglie wavelength? (iii) (a) If the particle is not free, but moves within an infinitely high potential well described by U(x) = 0 for 0 < x < L U(x) = for x 0 and x L adopt a trial solution to the Schrödinger equation of the form ψ(x) = A 1 e ikx + A 2 e ikx and derive an expression for the allowed set of energy levels in terms of Planck s constant, h, mass m, L and a series of quantum numbers, n = 1, 2, 3.... You will need to consider the boundary conditions at the boundaries of the potential well in order to derive the required equation. (b) Use the derived expression to calculate the energies, in Joules, of levels n = 1 and n = 2, assuming L = 5 10 10 m (a typical atomic dimension), and the mass to be that of an electron (9.1 10 31 kg). [9 marks] (iv) If the square well is not infinite as in: U(x) = 0 for 0 < x < L U(x) = U 0 for x 0 and x L E < U 0 where E is the energy of the particle discuss why a possible solution is to have a sinusoidal solution for ψ(x) inside the well, and an exponential solution outside it (where U(x) = U 0 ). [6 marks] [Total 25 marks] P.1.3 10 Please turn over

9. (i) Write down the Planck-Einstein energy equation and the De Broglie momentum equation, which describe the wave-like properties of all particles. [2 marks] (ii) Calculate the de Broglie wavelength of: (a) an electron (mass 9.11 10 31 kg) moving at a speed of 10 5 m s 1 (b) a bullet of mass 10 g, moving at 1 km s 1 (c) a human of mass 80 kg, running at a speed of 20 km hour 1 Compare the wavelengths obtained in each case with an appropriate typical length scale, which you should propose, and comment on the relevance of quantum effects. (iii) (a) Explain how the wave-particle duality of matter may be described conceptually by the use of the idea of a wave packet, meaning any group of individual waves that give rise to some degree of localisation of the wave in space (defined as a carrier wave vector, k c, and a range of wave vectors, Δk n, as in k n = k c + Δk n ). What implications are there from these ideas for Heisenberg s Uncertainty Principle for momentum-displacement? (b) Write down the dispersion relations for light waves and for particle waves. (c) Define the phase and group velocities, v p and v g, for a wave packet, and explain their physical significance. (d) Show that the phase and group velocities of the light waves are both equal to the speed of light, c, while for a de Broglie particle-wave the group velocity is twice the phase velocity. [11 marks] (iv) Consider a wave packet wave function defined as a set of N superposed travelling waves with complex amplitudes A n, having wave vectors given by N ψ(x, t) = A n exp [i (k n x ω n t)] (1) n=1 k n = k c + Δk n (2) and a corresponding set of angular frequencies, ω n (k n ) represented by a 2-term Taylor expansion of the dispersion relation for particle-waves (ignore higher terms): ω n (k n ) = ˉhk2 c 2m + dω Δk dk n (3) kc (a) Use equations (2) and (3) for k n and ω n to show by substitution that the general wave packet in equation (1) may be written in the form ψ(x, t) = f(k c x ω c t)g(x v g t) (4) P.1.3 11 [This question continues overleaf... ]

where f(x, t) = exp [i (k c x ω c t)] and N g(x, t) = A n exp [iδk n (x v g t)] n=1 (Hint: You will need to recognise that the terms in (3) relate to the carrier wave angular frequency, ω c, and the group velocity, v g ). (b) Briefly discuss the physical meaning of the f() and g() terms in equation (4) [7 marks] [Total 25 marks] P.1.3 12 End of examination paper