Spring07_questions

# Spring07_questions - Qualifying Exam Spring-2007 Classical...

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Qualifying Exam Spring-2007 Classical Mechanics Problem 1 A pencil of length L is held vertically with its point on a desk and then allowed to fall over. Assuming that the point does not slip, find the speed of the eraser just as it strikes the desk. Compare this with the speed that would result from free fall from a height equal to the length of the pencil. Qualifying Exam Spring-2007 Electrodynamics Problem 2 A static charge distribution produces a spherically radial electric field 2 exp( ) br E Ar r = r r where A and b > 0 are constants. 1) What is the charge density ρ (r) ? Make a sketch of the function. 2) What is the total charge Q? Qualifying Exam Spring-2007 Quantum Mechanics Problem 3 Consider two solutions to the one-dimensional time-independent Schrödinger equation with same energy E : 1 () x ψ and 2 x . (a) Prove that regardless of the potential function V ( x ), 12 21 dd C dx dx ψψ = where C is a constant. (b) Show that if 1 x and 2 x are bound state solutions, then C = 0. From this, show that γψ = for some constant γ , thus proving that there are no degenerate bound state solutions to the one-dimensional time-independent Schrödinger equation.

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(c) Why does this theorem fail for continuum eigenstates (i.e. unbound states)? Give a specific example of degenerate continuum eigenstates in one dimension. Qualifying Exam Spring 2007 Thermodynamics/Statistical Mechanics Problem 4 A certain amount of water of heat capacity C is at a temperature of 0 ° C. It is placed in contact with a heat reservoir at 100 ° C and the two come into thermal equilibrium. (a) What is the entropy change of the universe.
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Spring07_questions - Qualifying Exam Spring-2007 Classical...

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