Preliminary Examination - Day 2 Friday, May 11, 2018

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1 UNL - Department of Physics and Astronomy Preliminary Examination - Day Friday, May, 8 This test covers the topics of Thermodynamics and Statistical Mechanics (Topic ) and Quantum Mechanics (Topic ). Each topic has 4 A questions and 4 B questions. Work two problems from each group. Thus, you will work on a total of 8 questions today, 4 from each topic. Note: If you do more than two problems in a group, only the first two (in the order they appear in this handout) will be graded. For instance, if you do problems A, A3, and A4, only A and A3 will be graded. WRITE

Preliminary Examination - page Thermodynamics and Statistical Mechanics Group A - Answer only two Group A questions A A mass of.4 kg of a certain gas is contained within a piston-cylinder assembly.

2 Preliminary Examination - page Thermodynamics and Statistical Mechanics Group A - Answer only two Group A questions A A mass of.4 kg of a certain gas is contained within a piston-cylinder assembly. The gas undergoes a process for which the pressure-volume relationship is.5 pv = constant. The initial pressure is 3 kpa, the initial volume is. m 3, and the final volume is. m 3. The change in specific internal energy of the gas in the process is u u = 55 kj/kg. Determine the net heat transfer for the process. A Consider an engine working in a reversible cycle and using an ideal gas with constant heat capacity c as the working substance. The cycle consists of two processes at constant pressure, joined p by two adiabatics. Which temperature of T, T, T, a b c T is the highest and which is the lowest? Justify d your answer. A3 An automobile engine whose thermal efficiency is ε =.% operates at 95. cycles per second and does work at the rate of hp. How much heat does the engine absorb per cycle? ( hp = horsepower = 746 watts.) A4 A mixture of.78 kg of water and 6 g of ice at C is, in a reversible process, brought to a final equilibrium state where the water/ice ratio (by mass) is : at C. Calculate the entropy change of the system during this process.

3 Preliminary Examination - page 3 Thermodynamics and Statistical Mechanics Group B - Answer only two Group B questions B Given a round table, we randomly place three legs under the table. What is probability that the table will fall? B Two fluids, F and F, of fixed volumes and constant heat capacities C and C, are initially at temperatures T and T ( T > T ), respectively. They are adiabatically insulated from each other. A quasistatically acting Carnot engine uses F as heat source and F as heat sink, and acts between the systems until they reach a common temperature T. Find this T, and also find the total work done by the Carnot engine. B3 Consider the Joule-Thomson expansion in which a gas is allowed to flow slowly through a porous plug between two containers, which are otherwise isolated from each other and from their surroundings. The enthalpy is defined as H = U + pv. The temperature change of such an expansion is measured by the Joule-Thomson coefficient a. Show that dh = TdS + Vdp. V b. Subsequently, show that j ( Tα p ) V constant pressure is defined as α = p V T. p c. Finally, compute the coefficient j for an ideal gas. p T j p = where the coefficient of volume expansion at C H. B4 Consider a system of N particles with only three possible energy levels:, ε, and ε. The system occupies a fixed volume V and is in thermal equilibrium with a reservoir at temperature T. Ignore interactions between particles, and assume that Boltzmann statistics applies. a. What is the partition function for a single particle in the system? b. What is the average energy per particle? c. For each of the three energy levels, what is probability that it is occupied in the hightemperature limit, kt ε? B d. What is the average energy per particle for kt ε? B e. Find the heat capacity of the system, C, and give the limits for kt ε and for kt ε V B B.

4 Preliminary Examination - page 4 Quantum Mechanics Group A - Answer only two Group A questions A A An operator B is called anti-hermitian when anti-hermitian operators are purely imaginary. B = B. Show that the expectation values of A3 ϕi A4 Let the kets ( i =,,3,... ) be an orthonormal basis of some Hilbert space. We consider the operator P = Σ ϕ ϕ. i i i a. Show that P is idempotent, i.e. b. Is P Hermitian? P = P.

5 Preliminary Examination - page 5 Quantum Mechanics Group B - Answer only two Group B questions B Consider a charged oscillator, of positive charge q and mass m, which is subject to an oscillating electric field E cos( ω t). The particle s Hamiltonian is ˆ ˆ P H = + kxˆ + qe Xˆ cos( ωt ). m a. Calculate d Xˆ / dt, d P ˆ / dt, and d Hˆ / dt. b. Find X ˆ () t, the expectation value of ˆX as a function of time. B B3

6 B4 Preliminary Examination - page 6

7 Preliminary Examination - page 7 Physical Constants speed of light... 8 c =.998 m/s electrostatic constant... k = πε = 9 (4 ) m/f 34 3 Planck s constant... h = 6.66 J s electron mass... m el = 9.9 kg 34 Planck s constant / π... =.55 J s electron rest energy kev 3 Boltzmann constant... k =.38 J/ K Compton wavelength.. λ = hmc / =.46 pm B C el 9 elementary charge... e =.6 C proton mass... m = = electric permittivity... ε = F/m bohr... a 6 magnetic permeability... µ =.57 H/m hartree (= rydberg)... E molar gas constant... Avogadro constant... N R = 8.34 J / mol K gravitational constant... p kg 836 el m = / ke m =.59 Å = / m a = 7. ev h el G = m / kg s 3 = 6. mol hc... hc = 4 ev nm A el 3 Equations That May Be Helpful TRIGONOMETRY sin( α + β) = sinαcos β + cosαsin β sin( α β) = sinαcos β cosαsin β cos( α + β) = cosαcos β sinαsin β cos( α β) = cosαcos β + sinαsin β sin( θ) = sinθ cosθ cos( θ) = cos θ sin θ = sin θ = cos θ sinαsin β = cos( ) cos( ) α β α + β cosαcos β = cos( α β) + cos( α + β) sinαcos β = sin( α β) sin( α β) + + cosαsin β = sin( α + β) sin( α β) THERMODYNAMICS Partition function = Z= Σ Ei e β i = β Average energy = E ( lnz)

8 Preliminary Examination - page 8 Heat capacity = C V de = N dt Clausius theorem: N i= Q T i i, which becomes N i= Q T i i = for a reversible cyclic process of N steps. dp dt λ = TΔV For adiabatic processes in an ideal gas with constant heat capacity, pv γ = const. du = TdS pdv H = U + pv F = U TS G = F + pv Ω= F µ N δq S δq S S C = = T C = = T TdS = C dt + T dv dt T dt T V V p V V V p p T V V κ = α = V p V T T p X Y Z Triple product: = Y Z X Z X Y Maxwell s relations: SS = VV, SS = pp, TT = VV, TT = pp Data for water specific heat C = 486 J/(kg K) heat of fusion L = 334 kj/kg F heat of vaporization L = 56 kj/kg V QUANTUM MECHANICS e Ground-state wavefunction of the hydrogen atom: ψ () r = π Bohr radius, using m m, in which m is the electron mass. el el r/ a a / 3/, where a 4πε = me is the

9 Preliminary Examination - page 9 mk e ψ () r = R () ry () rˆ E = nlm nl lm n n r/ a R () r = e 3/ a r R () r = e a / 3/ 3 ( a ) r/a 4 Particle in one-dimensional, infinitely-deep box with walls at x = and x= a: / Stationary states ψ = (/ a) sin( nπx/ a), energy levels n Angular momentum: [ L, L ] = i L et cycl. x y z En = n π ma Ladder operators: L, m = ( + m+ )( m), m+ + L, m = ( + m)( m+ ), m Creation, annihilation operators: mω pˆ mω pˆ aˆ = xˆ i aˆ = xˆ + i mω mω ˆ = + + ˆ = a n n n a n n n Probability current density: Jx ( ) ψ ψ Im ψ = ψ ψ = ψ. mi x x m x

10 Preliminary Examination - page Hmag = γ SB Pauli matrices: i σ =, σ, σ x = y = i z Compton scattering: λ λ = λ ( cos θ ) C

11 Preliminary Examination - page

12 Preliminary Examination - page CARTESIAN AND SPHERICAL UNIT VECTORS xˆ = (sinθcos φ) rˆ+ (cosθcos φ) θˆ sin φˆ yˆ = (sinθsin φ) rˆ + (cosθsin φ) θˆ + cos φˆ zˆ = cosθ rˆ sin θˆ INTEGRALS + bx x e n bx dx = n dx = n b

Preliminary Examination - Day 1 Thursday, August 9, 2018

UNL - Department of Physics and Astronomy Preliminary Examination - Day Thursday, August 9, 8 This test covers the topics of Thermodynamics and Statistical Mechanics (Topic ) and Quantum Mechanics (Topic