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Department of Physics, Applied Physics, and Astronomy
Rensselaer Polytechnic Institute
110 Eighth Street, Troy, New York 12180-3590 USA

Telephone:
(518) 276-6310
Fax: (518) 276-6680

For questions about our graduate program, contact our Graduate Student Coordinator, Nicole McQuade.

Graduate Exam Handbooks >

Qualifying Exam Handbook

Qualifying Exam Committee
General Guidelines
Some Common Concerns Raised by Students
Recommended Textbooks
Syllabi
Past Exams

Qualifying Exam Committee

Gyorgy Korniss, Chair
Joel Giedt
John Schroeder
Ingrid Wilke
Shengbai Zhang

General Guidelines

The qualifying examination consists of two four-hour sessions covering materials at the level of advanced undergraduate courses.

  • Part I - Mechanics and Electrodynamics

    Intermediate Mechanics, similar to the topics presented in PHYS 2330, and Introduction to Electrodynamics, similar to the topics presented in PHYS 4210. Also some knowledge of special relativity at the undergraduate level.

  • Part II - Quantum Mechanics, Thermodynamics, and Introduction to Statistical Mechanics

    Introduction to Quantum Mechanics similar to topics presented in PHYS 4100. This part will also cover thermodynamics and introduction to statistical mechanics similar to the level of PHYS 4420.

The examination will be given twice a year, in August and in January.  Candidates for the Ph.D. degree must pass this examination within the 3rd semester of graduate study at Rensselaer. Students may take the examination in January (August) after the first semester Fall (Spring) semester. Those who do not pass may take the examination again in August (January) after the second semester. You have to take two parts simultaneously. The two parts may be passed separately. You get an extra try if you attempt to take the exam in August (January), just before the beginning of your first graduate Fall, (Spring) semester at Rensselaer. You can have at most three tries, counting this extra one. We encourage all incoming students to avail themselves of this "extra shot." It would save you a lot of time if you pass it at the first attempt on your arrival at Rensselaer. If you do not pass it, your exam results will allow you and your academic advisor to realize any deficiency in your undergraduate physics background and your advisor can advise you the proper courses to take. 

This examination normally consists of a number of problems, typically ten, distributed over four subjects, from which eight are required to be solved. A student is expected to make significant progress in a given problem to get a "pass" grade in that problem. Emphasis is placed on doing well on whatever is attempted, rather than accumulating points thinly over many problems.

This examination is the principal selection mode for entry into the doctoral program. The department expects that all students who pass this examination have the ability to continue successfully on to the Ph.D. degree. Of course students must also perform satisfactorily on other aspects of the program, which includes the appropriate course work (see graduate student handbook), the Candidacy Examination, and a research project leading to an acceptable thesis. Historically a large majority of our students have done so.

Copies of previous graduate level examinations are available from the secretary to the Graduate Program Committee.

Recommended Reference Textbooks

Undergraduate texts relevant to the qualifying examination:

  • Classical Dynamics of Particles and Systems, Fourth Edition, Jerry B. Marion and Stephen T. Thornton, HBJ Publishers.
  • Introduction to Electrodynamics, Third Edition, David J. Grifftiths, Prentice Hall, 1999.
    Or Classical Electromagnetic Radiation, Marion.
  • Introductory Quantum Mechanics, Liboff.
  • Classical and Statistical Thermodynamics, Ashley H. Carter, Prentice Hall 2001.
    Or, Introduction to Statistical Mechanics and Thermodynamics, Keith Stowe, Wiley & Sons, 1984.

Syllabi

Intermediate Mechanics

  • Review of Newtonian mechanics (kinematics and dynamics) of a single particle, Newton’s equations of motion in, two and three dimensions.
  • Conservation of linear and angular momenta, conservative forces, for a single particle.
  • Motion in a moving systemic coordinates, rotational dynamics, the Coriolis force, Euler angles and equation
  • Motion of a system of particles and the corresponding conservation laws, the center of mass and related topics.
  • The universal Law of Gravity. Motion in a central potential, motion in a 1/r2 force and Kepler’s laws.
  • Rotational kinematics and dynamics. The motion of rigid body and Euler’s equations of motion. Introduction to tensor analysis and the rotational Inertia tensor.
  • Calculus of variations as applied to mechanics.
  • Lagrange’s equations. Holonomic and non-Holonomic constraints. Independent generalized coordinates and generalized velocities.
    • The derivation of Lagrange’s equations from D’Alembert’s principle.
    • The derivation of Lagrange’s equations from a variation principle.
    • Cyclic coordinates, constant of motion
  • Hamilton’s equations of motion. Generalized coordinates and generalized momenta.
  • Normal frequencies, coupled oscillators, normal modes, small oscillations.
  • The special theory of relativity. The principle of relativity.
  • The constancy of the speed of light. Lorentz transformation. Relativistic kinematics and dynamics. Collisions.

Introduction to Electrodynamics

  • Review of elementary electrostatics. Coulomb’s law, Gausses law, the electrostatic field and the energy of this field.
  • Boundary value problems. Boundary value in two and three dimensions in rectangular coordinates, axially symmetric problems in spherical and cylindrical coordinates.
  • Method of images.
  • Multipole expansion for static fields.
  • Fields in matter, polarization and the displacement vector.
  • Magnetostatics. The Biot-Savart and Ampere laws. The static vector potential and its multipole expansion with the dipole term as the leading term. Currents and magnetic dipole moments.
  • Faraday’s law, Maxwell’s displacement current and electrodynamics.
  • Maxwell’s equations in free space and in linear homogeneous dielectric materials.
  • Vector and scalar potentials, the wave equation and gauge invariance.
  • Energy, momentum and angular momentum of electromagnetic fields, Maxwell’s stress tensor.
  • Plane waves, dipole radiation, Poynting vector.
  • The special theory of relativity. Lorentz transformation.

Introduction to Quantum Mechanics

  • Fundamental concepts:
    • Observable, Uncertainty relations and their implications.
    • Probabilistic interpretation. Correspondence principle, Complementarity Operators.
    • Matrix representation.
    • Dirac notation.
  • Schroedinger equation:
    • Wave functions and their time evolution.
    • Ehrnfest’s theorem.
    • Quantization of energy.
    • One dimensional problems.
      • Barrier reflection and tunneling.
      • One dimensional box and three dimensional box.
      • Quantum wells.
      • Harmonic oscillator. Stepping operators.
    • Three dimensional wave equation.
    • Central force problems.
      • Separation of variables.
      • Spherically symmetric solutions, spherical harmonics.
      • Quantization of angular momentum and vector model of angular momentum addition.
      • Spherical square well.
      • Spherical harmonic oscillator.
      • Hydrogen Atom.
      • Eigenfunction expansions.
  • Perturbation theory:
    • Time independent.
    • Second order, degeneracies.
    • Time dependent, Fermi’s golden rule.
  • Scattering theory:
    • Phase shift.
    • Partial wave analysis.
    • Born approximation.
  • Electron spin:
    • Stern - Gerlach experiment.
    • Spin - orbit coupling in atoms.
  • Identical particles and exchange symmetry:
    • Fermions, Bosons, Pauli exclusion principle.
  • Elementary applications:
    • Complex atoms, periodic table.
    • Homonuclear diatomic molecules.
    • Electronic, vibration, and rotation spectra.
    • Compton scattering, Photoelectric effect.

Thermodynamics and Introduction to Statistical Mechanics

Thermodynamics

  • Internal energy and equipartition.
    • Conservation of energy of a system of particles.
    • Work and heat.
    • 1st law of thermodynamics.
  • Entropy.
    • Entropy and 2nd law of thermodynamics.
    • Entropy and heat.
  • Interactions.
    • Thermal: Temperature & the zeroth law, absolute zero and the 3rd law, phase transitions, Clausius-Clapeyron equations, Ehrenfest equations, Heat capacity.
    • Mechanical: Work, thermal expansion & compressibilities.
    • Diffusive: Chemical potential, equilibrium conditions and the approach to equilibrium.
  • Constraints.
    • Ideal gas, real gas, liquids, solids.
    • Second law and third law constraints, Maxwell relations.
  • Processes (or called imposed constraints).
    • Isobaric, isothermal, and adiabatic processes.
    • Processes in terms of entropy.
    • Reversibility.
    • Nonequilibrium processes.
    • Applications to engines & refrigerators.
  • The equation of state.
    • Ideal gases & real gases.
    • Heat capacities of an ideal monoatomic gas & an ideal polyatomic gas.
  • Thermodynamic functions (Helmholtz free energy, Gibbs free energy, etc).
    • Thermodynamic identities.

Introduction to Statistical Mechanics

  • Fundamental concepts.
    • Phase space.
  • Classical.
    • Ensembles, probability in a microstate, applications.
    • Principle of equipartition of energy, applications.
    • The Maxwell-Boltzmann distribution, applications to the ideal gas and transport processes in gases.
  • Applications to Magnetism.
    • Paramagnetism, ferromagnetism, and phase transitions.
  • Quantum.
    • Indistinguishability.
    • Fermi-Dirac distribution, applications to the electron gas.
    • Bose-Einstein distribution, applications to photon gas, Bose-Einstein condensation.

Some Common Concerns Regarding the Qualifying Examination Raised by Students

The standard of the exam:
The qualifying exam committee follows strict instructions given to the qualifying exam committee by the entire Physics Faculty, which decides as a body the scope and syllabus of the exam. We follow advanced undergraduate level courses in intermediate mechanics, introduction to electrodynamics, introduction to quantum mechanics, thermodynamics and statistical mechanics.

What is the level of passing the written examination?
As a rule, passing in each part at least six problems, by scoring six points or better, out of ten points per problem, with a reasonable demonstration of proficiency in each sub-section, will guarantee success. Solutions of the latest examinations are available with the secretary to the Graduate Program Committee. These should give students examples of reasonable answers.

Grading:
Each member of the Qualifying Examination Committee grades an equitable number of problems. Grading is done anonymously, without the knowledge of the name of the particular student concerned.

What is the procedure by which students are evaluated?
After the Qualifying Examination Committee has graded the written examinations, the committee group’s examinations into tentative “Pass” and “Not Pass” categories. If the overall exam performance of a student who has not passed warrants it, the committee may recommend that additional information be gathered. Additional information will include course grades and instructor evaluations, and research and academic advisor evaluations. The committee will then make a recommendation to the faculty on whether to pass each student in this group. Finally, the faculty will discuss and vote on the committee’s recommendations. In general, only students who demonstrate good performance on the written examination in three out of four major areas will have additional information considered.

Appeal:
Appeals regarding grading of the examination are handled directly by the Qualifying Examination Committee. Please submit appeals to the Chair of the Qualifying Examination Committee, in writing, within a week of the publication of the results, after due consultation with the academic advisor. In case of an appeal, the entire Qualifying Examination Committee examines the contested question(s) to avoid any bias. Further appeals can be made to the Graduate Program Committee and eventually the Physics Department Chair, in exceptional circumstances.

Success rate:
There is no data available yet since this is a change from the past exams that tested graduate level materials. For the graduate level exams: Success rate for students has been as follows: a full 70% have successfully passed this examination since 1992. This rate has been fairly constant.

Deferment Procedure:
As a rule, we do not give deferment of the Qualifying Exam. In exceptional circumstances involving medical contingencies, the student in question, who cannot make the final regular attempt, may appeal to the Graduate Program Committee for a deferment. If a student enters the program with a deficient background, a deferred examination schedule can be formalized by applying immediately to the Graduate Program Committee. Part-time students may also apply for a deferred examination schedule.

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