Spring 1996 Part II (RPI)
Part II: Quantum Mechanics, Thermodynamics and Statistical Mechanics
INSTRUCTIONS FOR SUBMITTING ANSWER SHEETS FOR QUALIFYING EXAMINATION
Answer a total of 8 problems with at least 5 problems in quantum mechanics and 3 problems in thermodynamics and statistical mechaincs. Do not hand in more than 8 problems.
To pass the exam, 6 problems must be completed satisfactorily (with a score of 6/10 or better) with at least 4 in quantum mechanics and 2 in thermodynamics and statistical mechanics. The point value of each part of each problem is stated below the problem.
A. Quantum Mechanics
Problem 1:
Consider a physical system with a three-dimensional state space, spanned by the orthonormal basis formed by the three kets |u1>, |u2>, and |u3>. The Hamiltonian operator has the form
H=(hbar)ωM
and M is a (3x3)matrix with elements (sorry I can't make it pretty in this forum):
M13=M31=1
All other elements =0
A particular state vector at t=0 is given by
|ψ(t=0)>=(1/2)1/2|u1>+(i/2)|u2>+(1/2)|u3>.
a.) If a single measurement of the energy is made at t=0, what possible values of the energy can be found, and with what probabilities? What is the expectation value of H?
b.) Calculate the state vector at time t later. If the energy is measured at time t, what energy values can be obtained and with what probabilities?
[(3+2)+(2+2+1)]
Problem 2:
Consider a simple one-dimensional harmonic oscillator with the Hamiltonian H and its eigenvectors |φn> and eigenvalues (n+1/2)(hbar)ω, where n=0,1,2,3...
a.) Determine the eigenvectors of the annihilation operator a in terms of the eigenvectors of the Hamiltonian. (Denote the eigenvalue of a to be α (a complex number) and define a=(mω/2hbar)1/2x+ip/(2m(hbar)ω)1/2, with symbols having the usual meaning).
b.) When the system is in an eigenstate of a with an eigenvalue of α, what is the probability of finding the system in its ground state?
c.) What is the expectation value of H and of the position operator x when the system is in an eigenstate of a?
[4+2+(2+2)]
Problem 3:
Describe the difference between
a.) adiabatic and sudden approximations.
b.) scalar and pseudoscalar observables in quantum mechanics.
c.) the laboratory system and the center-of-momentum system in two-body scattering.
[3+4+3]
4. The operators J+ and J- are defined as
J±=Jx±iJy.
a.) Show that
J-|j,m>=N|j,m-1>,
where N is a normalization constant.
b.) Show that
J2=Jz2+J+J--Jz.
c.) Obtain an explicit expression for the normalization constant N in terms of j and m.
[4+2+4]
Problem 5:
The matrix for the Hamiltonian H is given by:
H11=E1
H22=E2
H12=H21=ΔE.
Assume that ΔE<<|E1-E2|.
a.) What is the leading order correction to the energy eigenvalues (of the Hamiltonian for which ΔE=0)?
b.) Obtain the eigenfunctions in the lowest order of perturbation theory.
c.) Obtain an expression for the exact eigenvalues and show that, in the limit of ΔE-->0, they are consistent with perturbation theory results.
[4+4+2]
Problem 6:
A beam of particles with uniform velocity v enters an interaction region, where some of them are absorbed. This is represented by a complex potential V1+iV2 (V1, V2 are real).
a.) Use the time-dependent Schrodinger equation to calculate the loss of flux due to absorption.
b.) Show that the cross section per atom for absorption is
σ=2V2/((hbar)Nv)
where N is the number of absorbing atoms per unit volume.
[8+2]
B. Thermodynamics and Statistical Mechanics
Problem 7:
Consider the van der Waal's equation of state
P=kBT/(v-b)-a/v2
where kB, a and b are constants, T is the temperature, P is the pressure and v is the molar volume.
a.) Give a physical interpretation of the constants a and b.
b.) Determine the volume, temperature and pressure of the critical point in terms of the given constants.
[3+7]
Problem 8:
Assume that particles of mass m at temperature T are in the one-dimensional potential V(x}=ε0|x/a|n, where ε0 and a are parameters with the units of energy and length respectively, and n is a constant.
a.) Calculate the partition function Z and the thermal energy E.
b.) Discuss your calculated results with respect to the specific heat cv of the above system.
[(3+3)+4]
Problem 9:
Consider a system of N distinguishable, non-interacting spins in a magnetic field H. Each spin has a magnetic moment of size μ, and each can point either parallel or antiparallel to the field. Thus, the energy of a particular state is
Σi=1N-niμH, with ni=±1
where niμ is the magnetic moment in the direction of the field. For the above described system:
a.) determine the average total magnetization
<M>=<Σi=1Nμni>
as a function of β=1/kBT, H and N (the total number of spins);
b.) similarly, determine <(δM)2>, where
δM=M-<M>,
and compare your result with the susceptibility given by
χ=(∂<M>/∂H)β,N;
c.) examine the behavior of <M> and <(δM)2> in the limit T-->0, and interpret your results.
[4+4+2]
Problem 10:
A nonideal gas has an equation of state P=P(n,T), where n is the particle density. The gas is at equilibrium in a uniform gravitational field g=-gk.
a.) Using hydrostatic arguments, obtain a differential equation for the density at height z.
b.) Show that for an ideal gas, the equation predicts the usual exponential density function n(Z)=N0e(-mgz/kBT).
[7+3]
edit: fixed subscript bracket
