Department of Physics PRELIMINARY EXAMINATION 2017 Part II. Long Questions/Answers

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1 Department of Physics PRELIMINARY EXAMINATION 2017 Part II. Long Questions/Answers Friday May 19th, 2017, 14-17h Examiners: Prof. K. Dasgupta, Prof. H. Guo, Prof. G. Gervais (Chair),Prof. D. Hanna, Prof. V. Kaspi, and Prof. P. Wiseman INSTRUCTIONS Answer 4 questions out of the choice of 8. You must attempt one problem in each category, i.e. one in classical mechanics/special relativity, one in quantum mechanics, one in electromagnetic theory and one in statistical mechanics/thermo. This is a closed book exam. Approved calculators may be used (non-programmable ones), though approximate numerical results are valid. If you attempt more than four questions, you should clearly mark which ones should be graded. DO NOT WRITE YOUR NAME; write ONLY you student ID on the exam booklet. Clearly indicate the question number next to each answer. This exam has 11 pages, including this title page.

2 2017 Prelim Long Questions/Answers 2 Statistical Mechanics/Thermodynamics Statistical Planck a) Show that in the large wavelength limit (λ >> hc/kt ), the Planck function, given by B ν (ν, T ) = 2hν3 1 c 2 e hν/kt 1, (1) where B ν is the spectral radiance at frequency ν c/λ and temperature T, can be reduced to a far simpler expression, which is known as the Rayleigh-Jeans function. b) Explain why the Rayleigh-Jeans function is classical, and verify the ultraviolet catastrophe, i.e. that it diverges for short λ. c) The Sun has T = 5800 K. At what approximate wavelength, and in what part of the EM spectrum, is the Sun s predicted Rayleigh-Jeans brightness roughly twice the actual brightness? 2

3 2017 Prelim Long Questions/Answers 3 Boltzmann extremization In the Grand Canonical Ensemble (GCE), the number of molecules (N) and amount of energy (E) are free to exchange between member systems in the ensemble while volume (V ), temperature (T ) and chemical potential (µ) are constant. The degeneracy of any macrostate (distribution) is given by the total number of microstates as a function of number and energy state occupancies: Ω(n N,j ) = N N! j n N,j! The equilibrium state is the distribution of molecules that has the maximum number of equivalent microstates, i.e. the maximum degeneracy Ω. This expression can be maximized by the method of Lagrange (undetermined) multipliers under three constraints. These are: the total number of systems in the ensemble (N = N j n N,j), the total energy (E t = N j n N,jE N,j ) and the total number of molecules (N t = N j n N,jN N,j ) and you can use them to find the that maximizes ln(ω). n N,j A) Show that ln(ω) is maximized by the expression n N,j = e α e βe N,j e γn N,j. Hint! Determine the first Lagrange α multiplier without evaluating the second and third Lagrange multipliers β and γ. B) Use your results from A) to show that the probability of a given macrostate is given by its Boltzmann factor normalized by the partition function: P N,k (V, β, γ) = where Ξ(V, βγ) = N k e βe N,k(V ) e γn. e βe N,k (V ) e γn Ξ(V,βγ) (2) 3

4 2017 Prelim Long Questions/Answers 4 Quantum Mechanics An analytically solvable quantum problem A particle with mass m moves in a one-dimensional potential V (x). Clearly it satisfies the Schrödinger equation Ĥ ψ = ( ˆT + ˆV ) ψ = E ψ where ˆT is the kinetic energy operator and ˆV the potential energy operator. We also know that the momentum eigenvalue equation is ˆp p = p p. (a) Determine (or write down) the p-basis p in the x-representation. (b) Calculate matrix elements of the potential operator in p-basis: V pq p ˆV q. Expanding V (x) in Taylor series of x and carry out the integration inside V pq, show that = V (ˆx)δ(p q), where ˆx = i h / p. V pq (c) Show that the energy eigenvalue equation in the momentum representation is: p 2 2m ψ(p) + V pq ψ(q)dq = Eψ(p) where ψ(p) is the wave function in p-representation, namely ψ(p) = p ψ. 4

5 2017 Prelim Long Questions/Answers 5 Massive degeneracy in a tube Two infinite two-dimensional plates with charge densities +Q each are separated by a distance L along the x-direction. These two plates are therefore extended along y and z-directions. We connect the two plates by a cylinder with base radius R such that R << L as shown in the figure below: +Q +Q L Assume that we have N massive charged fermions, each with mass m and charge q, that are constrained to move only on the surface of the cylinder. Calculate the Fermi energy of the system, ignoring mutual interactions between the particles. Observe also that the Gauss law constraint is not violated because the lines of force can escape to infinity. There might be other subtleties associated with these considerations, but you can ignore them for this problem. 5

6 2017 Prelim Long Questions/Answers 6 Electromagnetic Theory What if magnetic monopoles existed? In this problem, we are going to investigate magnetic monopoles. Assume the dielectric constant (ɛ), magnetic permeability (µ) and the speed (c) of light all equal one in the units used here. (a) Without magnetic monopoles, the Maxwell equations are: E = B t, and the continuity equation is D = ρ, B = 0, ρ t H = E t + J = 0. + J, Now, if magnetic monopoles exist in the universe, a magnetic charge density ρ m would exist. Modify the above Maxwell equations when this is the case. (b) To search for magnetic monopoles, people used the following idea. Suppose magnetic monopoles move slowly inside a circular wire C that goes through the middle of a loop (L, see the figure). Using one of the modified Maxwell equations of part (a), show the following relation is true: E dl = Φ I m t L where E is the electric field induced by the magnetic monopole current I m that is flowing in the circle, the contour integral is over the loop L, and Φ is the magnetic flux in the circle C. (c) Assume the loop L has resistance R and that Φ is extremely small and can be ignored. Show that the electric charge that goes around the loop L is Q = q m R, 6

7 2017 Prelim Long Questions/Answers 7 L q m C where q m is the magnetic charge. So by measuring Q, we can obtain q m. (Thus far, no reproducible q m has been detected.) 7

8 2017 Prelim Long Questions/Answers 8 Retarded fields The general expressions for the scalar and vector potentials are V (r, t) = 1 ρ(r,t r)dτ 4πɛ 0 and A(r, t) = µ 0 J(r,t r)dτ. r r 4π r r where t r is the retarded time. A long (effectively infinite) neutral wire on the z-axis has zero current for t < 0. At t = 0 a steady current, I 0, in the positive z direction is suddenly turned on. Figure 1: Charge, wire and retarded fields. (a) Consider a point at a distance s from the wire at z = 0. At what time do the electric and/or magnetic fields first become non-zero at this point? Hereafter call this time t s. (b) What is the value of the scalar potential V at position s at time t > t s? (c) What is the direction of the vector potential A at position s at time t > t s? 8

9 2017 Prelim Long Questions/Answers 9 (d) What is the direction of the electric field E at position s at time t > t s? (e) What is the direction of the magnetic field B at position s at time t > t s? (f) Write an integral expression for the magnitude of the vector potential A(s, t) at times t > t s. 9

2017 Prelim Long Questions/Answers 10 Classical Mechanics The particle in a parabolic bowl A particle of mass m is constrained to move under gravity without friction on the inside of a paraboloid of

10 2017 Prelim Long Questions/Answers 10 Classical Mechanics The particle in a parabolic bowl A particle of mass m is constrained to move under gravity without friction on the inside of a paraboloid of revolution whose axis is vertical (e.g. z = αr 2 ). (a) Show that the motion of the particle can be described by a single variable and derive the differential equation that, when solved, gives the time dependence of that variable. (You do not need to solve the differential equation.) (b) What is the condition on the particle s initial velocity to produce circular motion? 10

11 2017 Prelim Long Questions/Answers 11 The Galilean cart Consider a system where a spring is attached to a moving cart. The mass m moves freely, without friction, in the x-direction, and the cart is constrained to move along that axis at a constant speed v 0. We suppose the spring has a length equal to zero when at rest. We consider a set of coordinates that is fixed, and one that is outside of the cart. m k v 0 t a) Write down the Lagrangian for a set of coordinates outside of the cart. b) Calculate the Hamiltonian, and all canonical momenta. c) Is this Hamiltonian the energy of the system? Is it conserved? d) Redo the same problem but with a set of coordinates fixed to the moving cart. Make a Galilean transformation x x + v 0 t and derive the new Lagrangian and Hamiltonian in that case. e) Is this new Hamiltonian the energy of the system? Is it conserved? 11

Department of Physics PRELIMINARY EXAMINATION 2015 Part II. Long Questions

Department of Physics PRELIMINARY EXAMINATION 2015 Part II. Long Questions Friday May 15th, 2014, 14-17h Examiners: Prof. J. Cline, Prof. H. Guo, Prof. G. Gervais (Chair), and Prof. D. Hanna INSTRUCTIONS