UNL - Department of Physics and Astronomy Preliminary Examination - Day Thursday, August, 7 This test covers the topics of Quantum Mechanics (Topic ) and Electrodynamics (Topic ). Each topic has 4 A questions and 4 B questions. Work two prolems 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 prolems in a group, only the first two (in the order they appear in this handout) will e graded. For instance, if you do prolems A, A3, and A4, only A and A3 will e graded. WRITE YOUR ANSWERS ON ONE SIDE OF THE PAPER ONLY
Preliminary Examination - page Quantum Mechanics Group A - Answer only two Group A questions A The operator ˆR is defined y R ˆ ψ( x) = Re[ ψ( x)]. Is ˆR a linear operator? Explain. 3 A Find the energy levels of a spin s = particle whose Hamiltonian is given y ( ) ˆ α H Sˆ Sˆ Sˆ β Sˆ = + x y z z, where α and β are constants. A3 A4
Preliminary Examination - page 3 Quantum Mechanics Group B - Answer only two Group B questions B Consider a system which is initially in the normalized state ψθφ (, ) = Y ( θφ, ) + ay ( θφ, ) + Y ( θφ, ),,, 5 5 in which a is a positive real constant. a. Find a.. If L were measured, what values could one otain, and with what proailities? z We now measure L and find the value. z c. Calculate L and L. x y d. Calculate the uncertainties L and L, and their product L L. You may use the x y x y equality L = L without proving it first. x y B The Hamiltonian for a one-dimensional harmonic oscillator is ˆ pˆ H m x m = + ω ˆ. We write its energy eigenkets as n ( n =,,,... ) for energy E = ( n+ ) ω. a. Suppose the system is in the normalized state ϕ given y ϕ = c + c, and that the expectation value of the energy is known to e ω. What are c and c? i. Now choose c to e real and positive, ut let c have any phase: c = c e θ. Suppose further that not only is the expectation value of the energy known to e ω, ut the expectation value of x is also known: ϕ x ϕ =. Calculate the phase angle θ. mω c. Now suppose the system is in the state ϕ at time t =, i.e., ψ( t = ) = ϕ. Calculate ψ () t at some later time t. Use the values of c and c you found in parts a. and. d. Also calculate the expectation value of x as a function of time. With what angular frequency does it oscillate? Again, use the values of c and c you found in parts a. and. n
Preliminary Examination - page 4 B3 B4
Preliminary Examination - page 5 Electrodynamics Group A - Answer only two Group A questions A An isolated sphere of perfectly conducting material is surrounded y air. Though normally a good insulator, air reaks down (it ecomes conductive) for electric fields eyond 3. kv/mm (the so-called dielectric strength of air). The sphere s radius is 5. cm. What is the maximum amount of electrostatic energy the sphere can store efore reakdown occurs? Assume the electrostatic potential is zero at infinite distance from the sphere. A The diagram shows part of an electronic circuit. Calculate the potential at point P. 3 A3 If the electric field generated y a static point charge were proportional to /r, where r is the distance from the charge, would Gauss law still e correct? Justify your answer. A4 The diagram shows an infinitely long chain of resistors. What is the resistance etween points A and B? Hint: If the chain s length is increased y one unit, how does this resistance change?
Preliminary Examination - page 6 Electrodynamics Group B - Answer only two Group B questions B A flat square loop of wire of length a on each side carries a stationary current I. Calculate the magnitude of the magnetic field at the center of the square. B All space is filled with a material with uniform, fixed magnetization M, except for the region < z< a, in which there is vacuum. The magnetization is M= Mu, ˆ where û is a unit vector in the yz plane that makes an angle θ with the z-axis: uˆ = (sin θ) yˆ + (cos θ) z ˆ. Calculate the magnetic field B and the auxiliary field H everywhere. B3 A small all with mass m and electric charge +q is hung in a horizontal, uniform electric field E y a string of negligile mass. The all is raised to the position shown in the figure and then dropped from rest. At what angle θ will it come to rest again? B4 An infinitely large uniform sla of linear, isotropic dielectric material of permittivity ε is parallel to the xy plane. It is exposed to an external electric field E perpendicular to the sla (so, in the z direction). Find the polarization P inside the sla.
Preliminary Examination - page 7 Physical Constants speed of light... 8 c =.998 m/s electrostatic constant... k = πε = 9 (4 ) 8.988 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... 5. 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 = 8.854 F/m ohr... a 6 magnetic permeaility... µ =.57 H/m hartree (= ryderg)... E molar gas constant... Avogadro constant... N R = 8.34 J / mol K gravitational constant... p 7.673 kg 836 el m = / ke m =.59 Å = / m a = 7. ev h el G = 6.674 m / kg s 3 = 6. mol hc... hc = 4 ev nm A el 3 Equations That May Be Helpful TRIGONOMETRY sinαsin β = cos( α β) cos( α + β) cosαcos β = cos( ) cos( ) α β + α + β sinαcos β = sin( α β) sin( α β) + + cosαsin β = sin( α + β) sin( α β) 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 ψ () r = R () ry () rˆ nlm nl lm R () r = e 3/ a r/ a r R () r = e a E / 3/ 3 ( a ) n = n mk e 4 r/a
Preliminary Examination - page 8 Particle in one-dimensional, infinitely-deep ox 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 aˆ n = n+ n+ aˆ n = n n Hmag = γ SB Pauli matrices: i σ, σ =, x = σ y = i z Compton scattering: λ λ = λ ( cos θ ) C
Preliminary Examination - page 9 ELECTROSTATICS q encl E n ˆ da = E = V S ε a r r E dl = V( r ) V( r ) V() r = 4πε q( r ) r r Work done W= qe dl = qv ( ) V( a) Energy stored in elec. field: W= ε Edτ = Q C Relative permittivity: ε = + χ r e / V Bound charges ρ = P σ = P nˆ Capacitance in vacuum Parallel-plate: A C = ε d Spherical: a C = 4πε a Cylindrical: L C = πε ln( / a) (for a length L) MAGNETOSTATICS Relative permeaility: µ = + χ r m Lorentz Force: F= qe+ q( v B ) Current densities: I = J d A, I = K dl µ Id Rˆ Biot-Savart Law: Br () = 4π ( R is vector from source point to field point r ) R Infinitely long solenoid: B-field inside is B = µ ni (n is numer of turns per unit length) Ampere s law: B dl =µ I encl Magnetic dipole moment of a current distriution is given y Force on magnetic dipole: F= ( m B) Torque on magnetic dipole: τ= m B B-field of magnetic dipole: Br µ ( ) = 3( ˆˆ ) 3 4π r m= I da.
Preliminary Examination - page Bound currents J K = M = M nˆ Maxwell s Equations in vacuum ρ. E = Gauss Law ε. B = no magnetic charge 3. B E = t Faraday s Law 4. E B= µ J+ εµ t Ampere s Law with Maxwell s correction Maxwell s Equations in linear, isotropic, and homogeneous (LIH) media. D = ρ Gauss Law f. B = no magnetic charge B 3. E = Faraday s Law t D 4. H= J + Ampere s Law with Maxwell s correction f t Induction dφ B Alternative way of writing Faraday s Law: E d = dt Mutual and self inductance: Φ = M I, and M = M ; Φ= LI Energy stored in magnetic field: W = µ B dτ LI d = = A I V
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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 + x x e n x dx = π / n! dx = n+ / ( ) ( x + ) dx = ln x+ x + / ( x + ) dx = arctan( x/ ) 3/ x ( x + ) dx = x + x + arctan( x/ ) ( x + ) dx = x + 3 xdx = ln ( x + ) x + dx x = ln x( x + ) x + dx ax = ln ax a ax + ax = artanh a