Take A to be real with | A | | E 0 | . (a) (25 points) Find the exact eigenvalues of H . What are the corresponding normalized eigenstates in terms of the up state ( 1 0 ) and the down state ( 0 1 ) ? (b) (25 points) Next suppose that the molecule is placed in an electric field that distin- guishes the up and down states. The new Hamiltonian is H = H 0 + H 0 + H 00 , where H 00 = 1 0 0 2 with 1 6 = 2 . What are the exact energy eigenvalues of the new Hamiltonian H ? (c) (20 points) When | 1 | , | 2 | | A | , what answer does time-independent perturbation theory give for the energies of H to lowest non-vanishing order in i ? Compare your result to the exact answer. (d) (30 points) When | A | | 1 | , | 2 | , | 2 - 1 | , what answer does time-independent per- turbation theory give for the energies of H to lowest non-vanishing order in A ? Compare your result to the exact answer. 3
QUALIFYING EXAMINATION, Part 4 2:00 pm – 4:30 pm, Friday August 30, 2019 Attempt all parts of both problems. Please begin your answer to each problem on a separate sheet, write your 3 digit code and the problem number on each sheet, and then number and staple together the sheets for each problem. Each problem is worth 100 points; partial credit will be given. Calculators and cell phones may NOT be used. 1
Problem 1: Statistical Mechanics I Consider a non-interacting classical two-dimensional (2D) gas of N non-relativistic iden- tical particles confined in a harmonic trap with hard walls at r = R V trap ( r ) = 1 2 mω 2 r 2 for r < R ∞ for r ≥ R , where m is the particles mass, ω is the trap angular frequency and r is the radial coordi- nate. (a) (20 points) Show that the density distribution n ( r ) of this gas in thermal equilibrium at temperature T is n ( r ) = Ce - βV trap ( r ) , with β - 1 = k B T . Compute C such that n ( r ) is normalized as R n ( r )d 2 r = N . Sketch n ( r ) /n (0) as a function of r/R . Next, this gas is set to rotate at a fixed angular velocity Ω around the z axis, defined as perpendicular to the gas plane and intersecting it at the center of the trap r = 0. The ther- modynamics of a non-interacting gas in equilibrium in the rotating frame can be calculated by replacing the single-particle Hamiltonian in the lab frame h lab = p 2 / (2 m )+ V trap ( r ) by h rot = h lab - Ω · L , where Ω = Ω ˆ z ( ˆ z is the unit vector along the z axis) and L = r × p is the angular momentum of the particle (note that L is collinear with ˆ z here). (b) (25 points) Show that h rot can be written in the form h rot = 1 2 m ( p - m Ω × r ) 2 + V eff ( r ), where V eff ( r ) = 1 2 m ( ω 2 - Ω 2 ) r 2 for r < R (and ∞ otherwise), and interpret this result. (c) (20 points) Suppose the rotating gas is in equilibrium at a temperature T . Show that its density distribution is n Ω ( r ) = De - βV eff ( r ) , and calculate D to normalize this distribution as in (a). For simplicity, express your result in terms of the parameter α = m ( ω 2 - Ω 2 ) 2 k B T R 2 . Sketch n Ω ( r ) /n Ω (0) as a function of r/R for slow (Ω ω ), fast (Ω = ω ), and ultrafast rotation (Ω ω ). (d) (15 points) Show that the canonical partition function of the classical gas has the form Z = Z 0 (1 - e - α ) N where Z 0 is a proportionality factor independent of R (you do not need to calculate Z 0 ). (e) (20 points) Using the partition function in (d), calculate the force F = k B T ∂ log Z ∂R exerted by the gas on the disk walls, and thus the pressure P = F 2 πR . Express your results for PπR 2 Nk B T as a function of α .
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