Phys/Level /1/9/Semester, 009-10 (1 handout) UNIVERSITY OF SURREY FACULTY OF ENGINEERING AND PHYSICAL SCIENCES DEPARTMENT OF PHYSICS BSc and MPhys Undergraduate Programmes in Physics LEVEL HE PAPER 1 MATHEMATICAL, QUANTUM AND COMPUTATIONAL PHYSICS (PHY056) Time allowed: ½ hours ANSWER FOUR QUESTIONS: TWO FROM EACH SECTION USE SEPARATE ANSWER BOOKLETS FOR SECTIONS A AND B Internal Examiner(s): Dr David A Faux Dr Richard P L Sear External Examiner: Professor Richard Thompson The only calculators approved by the Physics Department are Casio : FX-8 Series, FX-83 Series, FX-85 ES, FX-115MS, 115W, 115S and FX-570W; Sharp : EL-531 LH ; Texas Instruments : TI-30X ; and Tandy : EC-4031, EC-403. The numbers at the end of each section of a question give an approximate indication of the marks available. Additional Materials: Department of Physics Formulae Booklet SEE NEXT PAGE
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Phys/Level /1/9/Semester, 009-10 (1 handout) 3 SECTION A MATHEMATICS IV Attempt TWO questions. Please use a SEPARATE answer booklet for Section A, writing clearly on the front which section you are answering. A1. The heat conduction equation in one dimension is T x 1 T = μ t, with µ a constant. Substitute into this equation the time t and space x dependent temperature distribution μ n π t nπ x T ( x, t) = Aexp cos, L L and show that this function T(x,t) is a solution of the heat conduction equation in one dimension. A, L and n are constants. Show that this function does not satisfy the two boundary conditions, T(x = 0,t) = 0, and T(x = L,t) = 0. How can this function be modified so that it does? 8 marks Find the general solution, x(t), to the second-order inhomogeneous ordinary differential equation d x dx + 5 = exp dt dt ( t). 1 marks SEE NEXT PAGE
Phys/Level /1/9/Semester, 009-10 (1 handout) 4 A. Find the gradient of the scalar function ( ax) sin( by) f ( x, y, z) = sin, where a and b are constants. Using this gradient, determine the rate of change of f along the direction of the vector u = (5,4,5). 7 marks Find the curl of the vector function ( x, e y, xy) v ( x, y, z) =. 4 marks (c) Find the general solution of the homogeneous ordinary differential equation d y dy + 3 + y = 0. dx dx Start with this general solution and impose the boundary conditions y( x = 0) = 1 dy ( x = 0) = dx and so obtain the particular solution that satisfies these boundary conditions. 9 marks
Phys/Level /1/9/Semester, 009-10 (1 handout) 5 A3. For a simple harmonic oscillator, the Schrödinger equation for the wavefunction ψ(z) is d ψ + dz 1 kz ψ = Eψ. The equation is written in dimensionless units. The eigenfunction for the ground state is the Gaussian function ( z / ) ψ ( z) = exp λ. By substituting the eigenfunction into the Schrödinger equation, find expressions for both λ and the eigenvalue, E. 11 marks In three dimensions and spherical polar coordinates, Laplace s equation for a spherically symmetric potential function u(r) is d u + dr r Find the general solution to this equation. du dr = 0. Two boundary conditions are imposed: 1) on the surface of a sphere of radius r = 0.1 m, the potential u(r = 0.1 m) = 5 mv ; ) as r tends to infinity, u tends to 0. Find the particular solution for u(r) outside the sphere. 9 marks SEE NEXT PAGE
Phys/Level /1/9/Semester, 009-10 (1 handout) 6 SECTION B QUANTUM PHYSICS Attempt TWO questions. Please use a SEPARATE answer booklet for Section B, writing clearly on the front which section you are answering. B1. A particle of mass m and energy E is constrained to move in one dimension and is subject to a constant potential V 0. (ii) Write down the Hamiltonian operator for this system. Show that the time-independent wave function 1 mark ( x) ψ = Ae ikx satisfies the momentum eigenvalue equation and that the eigenvalue, p, is equal to k. 3 marks (iii) Write down the time-dependent wave function Ψ(x, t) for this system and hence determine Ψ ( x, t). Provide a physical interpretation of ( x, t) Ψ. 3 marks
Phys/Level /1/9/Semester, 009-10 (1 handout) 7 A particle of mass m is confined to a one-dimensional potential well with infinitelyhigh walls such that ( ) V x = 0, x a = V0, a< x b, =, x > b where a and b are constants, and V 0 > 0 (as in the figure below). For the case E > V 0, the trial wave function ψ ( x) = Bcos k x, x a = Acos kx, a< x < b = 0, x > b is proposed where k, k, A and B are constants. Using the boundary condition at x = b, obtain an expression for k. 3 marks (ii) (iii) Using the appropriate boundary conditions at x = a, derive a relationship between k and k. How might this equation be solved for k? 5 marks Describe, with the aid of a diagram, the form of the wave functions for the case E < V 0. What would happen to the energy levels as a 0? 5 marks SEE NEXT PAGE
Phys/Level /1/9/Semester, 009-10 (1 handout) 8 B. In a system described as the one-dimensional equivalent of the hydrogen atom, a particle of mass m and of charge e is constrained to move along the x-axis such that ( ) V x =, x 0 e =, x > 0 4πε x 0 Write down the Hamiltonian for the region x > 0 and the energy eigenvalue equation satisfied by the wave function ψ(x). marks (ii) Explain why the energy eigenvalue equation is also referred to as the timeindependent Schrödinger equation. marks A trial wave function for the system described in is ψ where A and a 0 are constants. xa / 0 ( ) x = Axe, x> 0, Sketch the wave function for this system. marks (ii) (iii) Verify that ψ(x) is a solution to the energy eigenvalue equation provided that a0 = 4 πε0 / me. Find the energy eigenvalue corresponding to the state ψ(x). 8 marks Determine the probability of finding the particle in the region x < a 0 given that 3/ the normalisation constant A= a 0. 6 marks You may use the following standard integral: bx bx e x xe dx= x + b b b
Phys/Level /1/9/Semester, 009-10 (1 handout) 9 B3. Two physical observables are represented by the commuting operators Aˆ and Bˆ. Write down the mathematical expression satisfied by the operators. 1 mark (c) A particle of mass m is constrained to move along the x-axis under the influence of a potential V(x). Write down the Hamiltonian operator, Ĥ, for this system. Determine if Ĥ commutes with the operator representing position on the x-axis. Comment on the physical significance of this result. 5 marks A one-dimensional system consists of an electron of mass m and of charge e moving under the influence of the potential V(x) with a normalised energy eigenfunction given by ( ) 1/ 1/4 3/ a x / ψ x = π a xe where a is a constant. Determine and sketch the probability density function. Hence, or otherwise, find the expectation value for the position on the x-axis, x. Justify your answer. 4 marks (ii) Is x a conserved quantity? Justify your answer. marks (iii) Show that x = 3 a / and hence determine the uncertainty in the position of the electron. You may use the expression ( ) x x x Δ =. 3 marks (iv) (v) Estimate the uncertainty in the momentum of the electron in terms of a stating any assumption you make. 1 marks Assuming the electron is confined to a region typical of the spacing between atoms in a crystal, provide an estimate of the value of a. Hence, determine a numerical value for the uncertainty in the velocity of the electron. Does your result suggest that the electron is moving relativistically? 4 marks You may use the following standard definite integrals ax xe 4 ax xe π dx= a 3/ 3 π dx= 5/ 4a FINAL PAGE