Physics Qual Exam Problems at University of Washington, Study notes of Quantum Mechanics

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Physics Qual Exam Problems

spring 1996 – autumn 2011

University of Washington

September 13, 2012

Preface

This is a compendium of problems from past qualifying exams for physics graduate students at the University of Washington. This compendium covers the period preceding Autumn 2011 when the Department changed the format from a classic stand alone Qualifying Exam, (held late Summer and early Spring) into the current course integrated Masters Review Exam (MRE) format. The problems from the post Autumn 2011 period can be found in the separate MRE problems compendium. UW physics graduate students are strongly encouraged to study all the problems in these two compendia. Students should not be surprised to see a mix of new and old problems on future exams. The level of difficulty of the problems on the old Qualifying Exams and the new Masters Review Exams is the same. All problems from the Qualifying Exams that cover material beyond the first and second quarters of the quantum mechanics and electromagnetism courses have been removed from this compendium. Problems are grouped into four chapters:

1. Classical Mechanics
2. Electromagnetism
3. Quantum Mechanics
4. Thermodynamics and Statistical Mechanics

The actual exams for each section contained typically 2 problems. Their relative weight can be judged from the point assignments on the problems. The exam for each section had a maximum possible score of 100 points. Not all problems from all exams are listed in this compendium, because some are used several times or are very similar, while others do not apply to the current material anymore. Many faculty have contributed to the preparation of these problems, and many problems have received improvements from multiple people. Consequently, it is impossible to give individual attributions for problems. If you notice any typographical errors (no doubt there are still some), please send a note to the chair of the Exam Committee (currently Marcel den Nijs ) so improvements can be made.

Notation

Boldface symbols like r or k denote three-dimensional spatial vectors. Unit vectors pointing along coordinate axes are denoted as ˆex, ˆey, etc. Carets are sometimes (but not always) placed over quantum operators to distinguish them from c-numbers. Dots are sometimes used as shorthand for time derivatives, so f˙ ≡ df /dt. Implied summation conventions are occasionally employed. Physical constants appearing in various problems include:

c vacuum speed of light e electron charge me electron mass mp proton mass  0 vacuum permittivity μ 0 vacuum permeability Z 0 ≡ μ 0 c vacuum impedance h ≡ 2 πℏ Planck’s constant g Earth’s gravitational acceleration GN Newton gravitational constant kB Boltzmann’s constant

Trying to memorize SI values of all these constants is not recommended. It is much more helpful to remember useful combinations such as:

α ≡ e^2 /(4π 0 ℏc) ≈ 1 / 137 fine structure constant (300 K) kB ≈ 401 eV room temperature me c^2 ≈ 0 .5 MeV electron rest energy mp c^2 ≈ 1 GeV proton rest energy aB ≡ ℏ/(αmec) ≈ 0 .5 ˚A Bohr radius 1 2 α (^2) mec (^2) ≈ 13 .6 eV Rydberg energy ℏ c ≈ 200 MeV fm conversion constant 1 / 0 = μ 0 c^2 ≈ 1011 N m^2 /C^2 conversion constant mPl ≡

ℏc/GN ≈ 1019 GeV/c^2 ≈ 0. 2 μg Planck mass

Contents

• Preface
• Notation
• 1 Classical Mechanics
  • 1.1 Anisotropic Oscillators
  • 1.2 Bar on Springs
  • 1.3 Bar on String
  • 1.4 Bead on Rotating Hoop
  • 1.5 Bead on Rotating Wire
  • 1.6 Bead on Swinging Hoop
  • 1.7 Billiard Ball
  • 1.8 Block on Wedge
  • 1.9 Blocks on Slider
  • 1.10 Cart and Pendulum
  • 1.11 Catenary
  • 1.12 Central Force
  • 1.13 Central Potential (1)
  • 1.14 Central Potential (2)
  • 1.15 Central Potential (3)
  • 1.16 Central Potential (4)
  • 1.17 Closed Orbits
  • 1.18 Diatomic & Triatomic Vibrations
  • 1.19 Disk Dynamics
  • 1.20 Driven Masses (A)
  • 1.21 Driven Masses (B)
  • 1.22 Driven Masses (C)
  • 1.23 Dumbbell Dynamics CONTENTS CONTENTS
  • 1.24 Earth Wobble
  • 1.25 Elastic Rod
  • 1.26 Elliptical Orbit
  • 1.27 Foucault Pendulum (1)
  • 1.28 Foucault Pendulum
  • 1.29 Hanging Cable
  • 1.30 Inertia Tensor
  • 1.31 Inertially Coupled Pendula
  • 1.32 Linear Triatomic
  • 1.33 Mass on Disk
  • 1.34 Normal Modes and Driven Systems
  • 1.35 Orbit Perturbations
  • 1.36 Mass-spring system
  • 1.37 Particle Refraction
  • 1.38 Passing Stars
  • 1.39 Plumb Bob
  • 1.40 Rolling Cylinder
  • 1.41 Rotating Frame
  • 1.42 Rotating Pendulum
  • 1.43 Safe Driving
  • 1.44 Sliding dumbbell
  • 1.45 Sliding rod
  • 1.46 Spinning Disk
  • 1.47 Spring & Pulley
  • 1.48 Sweet Spot
  • 1.49 Symmetric Top
  • 1.50 Three Cubes
  • 1.51 Torsion Pendulum
  • 1.52 Vibrating Spring
  • 1.53 Waves on a String
• 2 Electromagnetism
  • 2.1 Blood Bath
  • 2.2 Charge Density Wave
• 2.3 Charged Pendulum CONTENTS CONTENTS
• 2.4 Conducting Sphere
• 2.5 Conductors in Dielectric
• 2.6 Currents and Forces
• 2.7 Current-Carrying Cable
• 2.8 Dipoles and Dielectrics
• 2.9 Electromagnet Hoist
• 2.10 Electrostatic Sheet
• 2.11 Electrostatic Cylinder (1)
• 2.12 Electrostatic Cylinder (2)
• 2.13 Electrostatic Shells
• 2.14 Electrostatic spheres
• 2.15 A Conducting Wedge inside a Cylinder
• 2.16 Energy and Momentum (1)
• 2.17 Energy and Momentum of Electromagnetic Fields
• 2.18 Faraday Effect
• 2.19 Gauging Away
• 2.20 Hall Probe
• 2.21 Ionospheric Waves
• 2.22 Inverse Electrostatics
• 2.23 Localized Current Density
• 2.24 Magnetic Torque
• 2.25 Magnetostatics
• 2.26 Maxwell’s Equations
• 2.27 Maxwell Stress Tensor
• 2.28 Metallic Conductors
• 2.29 Metallic Reflection
• 2.30 Multipoles and Capacitance
• 2.31 Planar Waveguide
• 2.32 Plasma Dispersion
• 2.33 Point Charge and Conducting Sphere
• 2.34 Rectangular Waveguide
• 2.35 Reflection and Refraction
• 2.36 Retarded Fields
  • 2.37 Retarded Potentials CONTENTS CONTENTS
  • 2.38 Slab Transmission
  • 2.39 Superconductors
  • 2.40 Small Loop, Big Loop
  • 2.41 Time Dependent Fields (1)
  • 2.42 Time Dependent Fields (2)
  • 2.43 Transverse Waves
  • 2.44 Two-dimensional Electrostatics
  • 2.45 Waveguides
• 3 Quantum Mechanics
  • 3.1 Angular Momentum & Angular Distributions
  • 3.2 Angular Momentum and Measurements
  • 3.3 Atom in a Harmonic Trap
  • 3.4 Bound States in One-dimensional Wells (2)
  • 3.5 Breaking Degeneracy
  • 3.6 Charged Oscillator
  • 3.7 Clebsch-Gordon Coefficients
  • 3.8 Commutator Consequences
  • 3.9 Cycling Around
  • 3.10 Density Matrices
  • 3.11 Delta-function Potential
  • 3.12 Discrete or Continuous
  • 3.13 Electron near Insulator
  • 3.14 Electron near Liquid Helium
  • 3.15 Electromagnetic Transitions
  • 3.16 Field Emission
  • 3.17 Harmonic Trap
  • 3.18 Hydrogen Atom in Electric Field
  • 3.19 Nonrelativistic Hydrogen Atom
  • 3.20 Hydrogenic States
  • 3.21 Hyperfine Splitting
  • 3.22 Neutron Spins
  • 3.23 Omega Baryons in a Harmonic Potential
  • 3.24 One-dimensional Schrodinger Equation (1)
  • 3.25 One-dimensional Schrodinger Equation (2) CONTENTS CONTENTS
  • 3.26 One-dimensional Schrodinger Equation (3)
  • 3.27 Oscillator Excitement
  • 3.28 Perturbation Theory
  • 3.29 Potential Step
  • 3.30 Probability and Measurement
  • 3.31 Scattering off Potential Wells
  • 3.32 Spin Flip
  • 3.33 Spin Measurements
  • 3.34 Spin-1/2 Systems
  • 3.35 Spin-Orbit Interaction
  • 3.36 Spin Projections and Spin Precession
  • 3.37 Stark Shift
  • 3.38 Superposition
  • 3.39 Symmetric Top
  • 3.40 Three Spins (1)
  • 3.41 Three Spins (2)
  • 3.42 Three Spins (3)
  • 3.43 Time Evolution
  • 3.44 Time-Reversal
  • 3.45 Toy Nuclei
  • 3.46 Transmission and Reflection (A)
  • 3.47 Transmission and Reflection (B)
  • 3.48 Two-dimensional Oscillator
  • 3.49 Two Spins
  • 3.50 Unreal Helium
• 4 Thermodynamics and Statistical Mechanics
  • 4.1 Atmospheric Physics
  • 4.2 Basic Kinetic Theory (this problem has 2 pages)
  • 4.3 Bose-Einstein Condensation
  • 4.4 Classical and Quantum Statistics
  • 4.5 Clausius-Clapeyron Relation
  • 4.6 Cold Copper
  • 4.7 Free Expansion
• 4.8 Einstein solids CONTENTS CONTENTS
• 4.9 Entropy Change
• 4.10 Equipartition
• 4.11 Excitable Particles
• 4.12 Gambling
• 4.13 Hard Rods
• 4.14 Harmonic Oscillator
• 4.15 Helmholtz Free Energy
• 4.16 Ideal Fermions in any Dimension
• 4.17 Ideal Gas Entropy
• 4.18 Ideal Gas Zap
• 4.19 Ideal Paramagnet
• 4.20 Joule Free Expansion
• 4.21 Free Expansion
• 4.22 Landau Diamagnetism
• 4.23 Magnetic Cooling
• 4.24 Magnetic One-dimensional Chain
• 4.25 One-dimensional Conductors
• 4.26 Maxwell-Boltzmann Distribution
• 4.27 Mixture of Isotopes
• 4.28 Natural Gas Silo
• 4.29 Number Fluctuations
• 4.30 Particles in Harmonic Potential
• 4.31 Pendulum Oscillations
• 4.32 Photon Gas
• 4.33 Photons and Radiation Pressure
• 4.34 Quantum Particles in Harmonic Oscillator
• 4.35 Quantum Statistics
• 4.36 Refrigerating Ideal Gases
• 4.37 Rotating Cylinder
• 4.38 Rubber Band versus Gas
• 4.39 Stability and Convexity
• 4.40 Thermal Zipper
• 4.41 Thermodynamic Relations

CONTENTS CONTENTS

4.42 Two-dimensional Bosons.............................. 204 4.43 Two Levels..................................... 205 4.44 Two Levels and Beyond.............................. 206 4.45 Van der Waals Gas................................. 207 4.46 White Dwarfs.................................... 208

1.2. BAR ON SPRINGS CM

1.2 Bar on Springs

A rigid uniform bar of mass M and length L is supported in equilibrium in a horizontal position by two massless springs attached at each end.

The identical springs have the force constant k. The motion of the center of mass is constrained to move parallel to the vertical x-axis. Furthermore the motion of the bar is constrained to lie in the xz-plane.

A. [ 5 points] Show that the moment of inertia for a bar about the y axis through its center of mass is M L^2 /12. B. [ 15 points] Construct the Lagrangian for this bar-spring arrangement assuming only small deviations from equilibrium. C. [ 15 points] Calculate the vibration frequencies of the normal modes for small amplitude oscillations. D. [ 5 points] Describe the normal modes of oscillation.

2002au 12

1.3. BAR ON STRING CM

1.3 Bar on String

φ

θ

L

32 L

x

y

A thin uniform bar of mass M and length 32 L is suspended by a string of length L and negligible mass, as shown in the figure. [Note: The moment of inertia of a thin uniform bar of length l and mass m about its center of mass, perpendicular to its length is 121 ml^2 .]

A. [ 8 points] In terms of the variables θ and φ shown in the figure, what is the position and velocity of the center of mass of the bar in the xy-plane? B. [ 8 points] Write the Lagrangian for arbitrary angles θ and φ, and write the Lagrangian appropriate for small oscillations. C. [ 7 points] Find the Euler-Lagrange equations and show that the equations of motion for the angles θ and φ are Lθ¨ + L φ¨ + gθ = 0, L φ¨ + 34 Lθ¨ + gφ = 0.

D. [ 8 points] Write down the form of the normal modes of the system and solve for the frequencies of the normal modes. E. [ 10 points] Describe, both quantitatively and qualitatively, the motion of each normal mode.

Consider the situation where initially the system is at rest with θ = φ = 0. Starting at time t = 0, a constant force of magnitude F is applied horizontally to the bottom of the rod.

F. [ 7 points] How are the equations of motion that you found in part C modified by the force. G. [ 6 points] After a very short time ∆t, how are θ and φ related?

2003au 13

1.5. BEAD ON ROTATING WIRE CM

1.5 Bead on Rotating Wire

A bead of mass m slides without friction along a straight wire at an angle θ 0 from vertical that is rotating with a constant angular velocity Ω about a vertical axis. A downward vertical gravitational force mg acts on the bead.

A. [ 8 points] Show that the Lagrangian for the bead, using as a generalized coordinate the displacement s measured along the wire from the point of intersection with the rotation axis, is L = 12 m

s˙^2 + s^2 Ω^2 sin^2 θ 0

− mgs cos θ 0. B. [ 8 points] Obtain the equations of motion from the Lagrangian, and show that the condition for equilibrium at constant position on the wire is s = s 0 ≡ g cos θ 0 /(Ω sin θ 0 )^2. C. [ 10 points] Derive the above result for s 0 by directly applying Newton’s second law of motion to the bead. D. [ 10 points] Discuss the stability of this orbit against small displacements along the wire by finding an equation for the deviation η(t) ≡ s(t) − s 0. E. [ 8 points] Find the constraint force which keeps the bead moving with uniform angu- lar velocity in the φˆ-direction as the displacement s varies. You may use Lagrangian methods or Newton’s Second Law. F. [ 9 points] Find H, the Hamiltonian of the system, in terms of a suitable coordinate and momentum. G. [ 12 points] Show (i) whether H is conserved, (ii) whether H is equal to the sum of the kinetic and potential energies, and if not, (iii) whether the energy of the bead is conserved. Explain the physics behind your answers.

2001sp 15

1.6. BEAD ON SWINGING HOOP CM

1.6 Bead on Swinging Hoop

A thin hoop of radius R and mass M is free to oscillate in its plane around a fixed point P. On the hoop there is a point mass, also M , which can slide freely along the hoop. The system is in a uniform gravitational field g.

A. [ 20 points] Introduce appropriate coordinates describing the combined motion of the hoop and point mass, showing them on the diagram above. How many unconstrained coordinates are required? B. [ 50 points] Consider small oscillations. Derive the Lagrangian, and the Lagrange equa- tions. Find the normal mode eigenfrequencies. C. [ 30 points] Find the normal mode eigenfunctions and sketch the motion associated with each eigenfunction.

1999sp 16

1.8. BLOCK ON WEDGE CM

1.8 Block on Wedge

A wedge of mass M sits on a frictionless table. A block of mass m slides on the frictionless slope of the wedge. The angle of the wedge with respect to the table is θ. Take x positive to the right. Let the coordinates for the block be (x 1 , y 1 ), and those of the point of the wedge (x 2 , 0).

A. [ 8 points] Write the constraint equation for the block sliding on the wedge and the Lagrangian for the block and wedge. B. [ 22 points] Derive the equations of motion for the block and wedge, using the method of Lagrange Multipliers to incorporate the constraint.

2002sp,2007sp 18

1.9. BLOCKS ON SLIDER CM

1.9 Blocks on Slider

A platform of mass M sits on a frictionless table. Two identical blocks of mass m are attached with identical springs to a post fixed to the platform. The springs are massless and have force constant k. The blocks move on the frictionless surface of the platform and are constrained to move along the x axis (parallel to the table surface and in the plane of this paper).

A. [ 10 points] Give the Lagrangian of the system of masses and springs. B. [ 25 points] Calculate the normal frequencies of the system. C. [ 5 points] Describe the normal modes of vibration corresponding to these frequencies.

2002sp,2007sp 19