Qualifying Exam. Aug Part II. Please use blank paper for your work do not write on problems sheets!

Size: px

Start display at page:

Download "Qualifying Exam. Aug Part II. Please use blank paper for your work do not write on problems sheets!"

Joshua Adams
5 years ago
Views:

1 Qualifying Exam Aug Part II Please use blank paper for your work do not write on problems sheets! Solve only one problem from each of the four sections Mechanics, Quantum Mechanics, Statistical Physics and Electricity and Magnetism

2 I. Mechanics Problem M1 A simple pendulum of mass m!, in the uniform gravitational field g, has a mass m! at the point of support. Mass m! can move on a horizontal line x lying in the plane in which m! moves. a) Find the Lagrangian of the system. b) Find the excludable (cyclic) variable, the corresponding conserved quantity (constant of motion) and the physical meaning of the latter. Explain why this quantity is conserved. Hint: You can also find the constant of motion directly from the Lagrange equations.) c) If initially both m! and m! are at rest on the horizontal line x (φ = π 2), what is value of the above constant? d) Calculate the x coordinate of the center of mass of the system. What does the result of part c) tell you about the position of the center of mass as a function of time? Hint: differentiate the x coordinate of the center of mass on time or integrate the equation of part c). e) Find the energy of the system and the frequency of small oscillation. Interpret your result for the latter in the limit m! Hint: Express the x and y coordinates of mass m! in terms of φ by using the results of parts c) and d).

3 Problem M2 A. Prove that the angular momentum L = mr v is conserved in a central field V r. Proceed to show that the motion takes place in the plane perpendicular to L. (Hint: evaluate L r.) B. Consider now a particle of mass m moving in the Kepler field, V r = α r a. Write the expression for the energy E = 1 2 mv! α r and the angular momentum L in polar coordinates b. Eliminate the angular variable using the expression for L to write E in terms of the radial variable and its time derivative. Find the effective potential energy V!"" r for the radial motion and sketch it. c. Using V!"" r, find the radius r! of the circular orbit for a given E and show its location on your sketch of V!"" r. (Hint: relate V!"" r! to E.) d. Substitute r! into the expression for L to find the angular frequency ω! of the motion on a circle around the center.! e. Expand V!"" r around r! up to quadratic terms r r! to show that E from part c is that of the harmonic oscillator. Find the frequency ω! of the radial oscillations and the ratio ω! ω!.

4 II. Quantum Mechanics Problem Q1 Consider a quantum one- dimensional harmonic oscillator with the Hamiltonian H = ħ! 2m d! dx! + mω! x! 2 A. Find the eigenfunctions of the ground state and the first excited state and their energies. You may want to take the following steps: a. Using the oscillator length a = ħ mω as the unit of length and ħω as the unit of energy, affect a rescaling x xa, E Eħω to rewrite the stationary Schrödinger equation in the dimensionless form. b. Find the asymptotic solution ψ!"#$ x of this equation for x ±. c. Look for a full solution in the form. ψ x = ψ!"#$ x y x d. Solve for y x (Hint: in general, one uses a series solution, but since here we are looking just for the first two eigenfunctions, the method can be greatly simplified.)! e. Normalize the eigenfunctions. (Hint: exp αx! dx!! B. Find the time- dependent oscillator wavefunction Ψ x, t, if = π α). Ψ x, 0 = 2 3 π!! x 1 exp x! 2

5 Problem Q2 Below n = sin θ cos φ, sin θ sin φ, cos θ is the unit vector defining a direction in space and σ = σ! ı + σ! ȷ + σ! k is the vector operator, where are the Pauli matrices. σ! = σ! = 0 i i 0 σ! = A. Solve the eigenvalue/eigenfunctions problem for s = 1 2 to find the spin functions ψ!! (i = x, y, z), of the states with definite projections of the spin on x, y, and z axes. B. Find the operator s! of the spin projection in a direction of the unit vector n and express it in the matrix form. C. Find the mean value s! of the spin projection on n for a particle in a state with a definite s!. (Hint: Consider s! = 1 2 and s! = 1 2 separately and then unify your answer in terms of s! ). D. Find the probabilities of spin projections s! = 1 2 and s! = 1 2 for a particle in a state with a definite s!. E. Show that, in a state of two s = 1 2 particles with a definite value of the total spin, the operator σ! σ! also takes a definite value.

6 III. Statistical Physics Problem S1 Evaluate specific heat per particle of mass m in an infinite one- dimensional well of size L in the limit of high and low temperatures T. Please follow the following steps: A. Find the energy levels in the well. B. Write down the partition function Z. You may use the notation β = 1 k! T and the following dimensionless parameter: α = ħ! π! 2mL! k! T = ħ! π! β 2mL! C. Determine the condition of low and high temperature in terms of parameter α. D. For low temperature, limit the partition function to the first two terms (why can we do this?) and then expand ln Z. E. Evaluate free energy, entropy and energy in this limit. F. Evaluate the specific heat in this limit. G. For high temperature, convert the sum on n (quantum numbers) into an integral on x, where x = nα. H. Evaluate free energy, entropy and energy in this limit. I. Evaluate the specific heat in this limit.

7 Problem S2 Consider a completely degenerate (at a temperature of absolute zero) electron gas of N electrons. A. For a one- dimensional gas, in a metal of length L, evaluate a. The Fermi momentum p! and the Fermi energy ε!. (Hint: be careful with the limits of integration.) b. The total energy of the gas and the energy of the gas per electron; express the latter in terms of ε!. B. For a two- dimensional gas, in a metal of area A, evaluate a. The Fermi momentum p! and the Fermi energy ε!. (Hint: be careful with the limits of integration.) b. The total energy of the gas and the energy of the gas per electron; express the latter in terms of ε!.

8 IV. Electricity and Magnetism Problem E1 The circular plates of a parallel plate capacitor in vacuum are separated by a small distance d r, the radius of the plates. The capacitor is being charged by a current i. A. Calculate the electrostatic energy W! stored in the capacitor in terms of the plate charge Q, d and r. B. Calculate the rate of increase of W!,!"!!, in terms of the current i and the voltage V between the plates. (Hint: i =!"!!"!", Q = CV.) C. Calculate the magnetic field at the edges of the capacitor in terms of i and r. D. Calculate the Poynting vector at the edges of the capacitor E. Calculate the energy flux entering the capacitor. Compare you result with B.

9 Problem E2 A conducting sphere of charge q and radius a is placed in a uniform electric field E!. A. Calculate the potential inside and outside the sphere B. Calculate the charge density on the surface of the sphere C. Calculate the sphere s dipole moment D. Give interpretation of each of the terms for the outside potential E. Calculate the electric field inside and outside the sphere

Joint Entrance Examination for Postgraduate Courses in Physics EUF

Joint Entrance Examination for Postgraduate Courses in Physics EUF First Semester/01 Part 1 4 Oct 011 Instructions: DO NOT WRITE YOUR NAME ON THE TEST. It should be identified only by your candidate number