Preliminary Examination - Day 1 Thursday, August 9, 2018

Size: px

Start display at page:

Download "Preliminary Examination - Day 1 Thursday, August 9, 2018"

Kenneth Young
5 years ago
Views:

1 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 ). 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 YOUR ANSWERS ON ONE SIDE OF THE PAPER ONLY

2 Preliminary Examination - page Thermodynamics and Statistical Mechanics Group A - Answer only two Group A questions A Derive the expression for the efficiency, defined as the total work done over the total heat supplied, for a Carnot cycle which uses a monoatomic ideal gas as an operating substance. Use the equation of state for the gas PPPP nnnnnn and the internal energy UU 3 nnnnnn. A Prove that the CC pp for an ideal gas is independent of pressure. Reminder: heat capacity at constant pressure can be defined as CC pp ( ) pp. A3 The internal energy for kg of a certain gas, in joules, is given by UU.7 TT + CC where T is the gas temperature in kelvin, and C is a constant. The gas is heated in a rigid container (i.e. at constant volume) from a temperature of 4 C to 36 C. Compute the amount of work and heat flow into the system. A4 A large number of non-interacting particles is in equilibrium with a thermal bath of temperature 3 K. The particles have only three energy levels: E mev, E 3 mev, and E 4 mev. Calculate the average energy of a particle. 3

3 Preliminary Examination - page 3 Thermodynamics and Statistical Mechanics Group B - Answer only two Group B questions B Consider mixing g of water at 3 K with 5 g of water at 4 K. Calculate the final equilibrium temperature if the specific heat c of water per gram is cal/g/k. Calculate the change in entropy for this irreversible process. B A two-dimensional vector B of constant length B B is equally likely to point in any direction specified by the angle θθ. What is the probability that the xx-component of this vector lies between BB xx and BB xx + ddbb xx? B3 Show that the work done by a gas under arbitrary changes of temperature and pressure can be determined in terms of the coefficient of volume expansion at constant pressure αα pp and the isothermal compressibility coefficient κκ TT. As a corollary, show that for an isochoric (constant volume) process αα pp VV κκ TT Verify this for an ideal gas. Reminder: the involved coefficients are defined as αα pp VV and κκ TT pp VV. TT

4 Preliminary Examination - page 4 B4 Consider a paramagnetic material whose magnetic particles have angular momentum quantum number J which is an odd multiple of. The z-component can take J + values ( Jz J, J +, J +,..., J ). This leads to J + allowed values of the z-component of a particle s magnetic moment: µ z Jµ, ( J + ) µ, ( J + ) µ,..., J µ, where µ is some unit magnetic moment*. The energy of the magnetic moment in a magnetic field pointing in the +z direction is µ z B. *Not to be confused with the magnetic permeability for which the same symbol is used! a. Derive an expression for the partition function Z of a single magnetic particle in a magnetic field B pointing in the +z direction. Write your answer in terms of hyperbolic sine functions. You may find it convenient to use the variable b µ B β, where β / kt B as usual. b. Derive an expression for the average energy of the particle in part (a). Give your answer in terms of the hyperbolic cotangent function.

5 Preliminary Examination - page 5 Quantum Mechanics Group A - Answer only two Group A questions A A

6 Preliminary Examination - page 6 A3 A wavefunction in one dimension is given by C for a< x< 3a ψ ( x) elsewhere where C and a are positive constants. Calculate the expectation value of the parity operator. A4 The spherical harmonics are orthonormal; we have Y ( θφ, ) Y (, ) d, m θφ δ δ, m Ω mm where dω is an infinitesimal amount of solid angle, and the integral is taken over all solid angle. Use this expression to demonstrate that Y and Y are orthogonal.,,

7 Preliminary Examination - page 7 Quantum Mechanics Group B - Answer only two Group B questions B B NOTE: In this problem, we encounter infinitely large matrices, We will write these by only specifying the 4 by 4 block in???????? the upper left corner, as in????. For instance, the identity operator is written as ˆ I.???? The stationary states of the harmonic oscillator are defined by Hˆ n ( n+ ) ω n. β i The annihilation operator â of the harmonic oscillator is defined by aˆ xˆ + pˆ mω (with β mω / ). The operation of the annihilation operator is aˆ n n n. Thus, in the n basis, the annihilation operator s matrix is ˆ a 3 T a. Explain why aˆ aˆ, where T means matrix transposition. b. Find the matrix for â. c. Find the matrix for ˆx. d. Find the matrix for ˆp. e. Find the matrix for xp ˆˆ. T f. Explain why px ˆˆ [( xp ˆˆ) ]*, where T means matrix transposition. g. Find the matrix for px ˆˆ. h. Find the matrix for [ xp ˆ, ˆ] and comment on your answer.

8 Preliminary Examination - page 8 B3 B4 Consider a two-state quantum system. In the orthonormal and complete set of basis kets and, the Hamiltonian operator for the system is represented by ( ω > ): Hˆ ω 3 ω 3 ω + ω. Let us consider another orthonormal and complete basis, α and β, such that Ĥ α E α and Ĥ β E β (with E < E ). Let the action of some operator Â on the basis kets α and β be given by Aˆ α ia β and Aˆ β ia α 3 a β, where a is real and a >. a. Show that Â is Hermitian, and find its eigenvalues. Answer the next two independent parts based on the information given above: PART I - Suppose an Â -measurement is carried out at time t on an arbitrary state, and the largest possible value is obtained. b. Calculate the probability Pt () that another measurement made at some later time t will yield the same value as the one measured at t. c. Calculate the time dependence of the expectation value Â. Plot A ˆ () t as a function of time. What is the minimum value of Â? At what time is it first achieved? PART II - Suppose that the average value obtained from a large number of Â -measurements on identical quantum states at a given time is a /4. d. Construct the most general normalized ket (just before the Â -measurement) for the system consistent with this information. Express your answer as C α + D β.

9 Preliminary Examination - page 9 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 J s electron rest energy kev 3 Boltzmann constant... k.38 J/ K Compton wavelength.. B λ hmc /.46 pm 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)

10 Preliminary Examination - page HYPERBOLIC FUNCTIONS x x e e sinh( x) x x e + e cosh( x) sinh( x) tanh( x) cosh( x) coth( x) tanh( x) THERMODYNAMICS Partition function Z Σ Ei e β i β Average energy E ( lnz) 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

11 Preliminary Examination - page 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 el el m is the electron mass. r/ a a / 3/, where a 4πε me is the 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

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

13 Preliminary Examination - page 3

Preliminary Examination - page 4 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

14 Preliminary Examination - page 4 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 π /b n! dx b n+ / ( ) ( x + b ) dx ln x+ x + b / ( x + b ) dx arctan( x/ b) b 3/ x ( x + b ) dx b x + b bx + arctan( x/ b) ( x + b ) dx x + b 3 b xdx ln ( x + b ) x + b dx x ln x( x + b ) b x + b dx ax b ln ax b ab ax + b ax artanh ab b

Preliminary Examination - Day 2 Friday, May 11, 2018

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