APS Graduate Exam key terms

A method of checking equations and deriving relationships by requiring that both sides have identical physical units. It can predict how a quantity scales with parameters, often up to a dimensionless constant.

A scalar function equal to kinetic energy minus potential energy for a system. Equations of motion follow from requiring the action integral of this function to be stationary.

A function, usually total energy expressed in coordinates and conjugate momenta, that generates time evolution. Its canonical equations give the system's dynamics in phase space.

The time integral of the Lagrangian along a path between two configurations. Physical trajectories are those for which this quantity is stationary under small variations.

Any minimal set of independent variables that fully specify a system's configuration, chosen to respect constraints. They need not be Cartesian positions and reduce the number of equations needed.

A force that enforces a geometric or kinematic restriction, such as a normal force or tension, and does no virtual work along allowed displacements. Its magnitude is determined by the motion, not given in advance.

An antisymmetric operation on two phase-space functions built from partial derivatives with respect to coordinates and momenta. It encodes dynamics: the bracket of a quantity with the Hamiltonian gives its time derivative.

A change of phase-space variables that preserves the form of Hamilton's equations. Such transformations are generated by functions of old and new coordinates and momenta.

An independent oscillatory pattern of a coupled linear system in which all parts move at a single common frequency. Any small motion is a superposition of these patterns.

A result stating that every continuous symmetry of an action corresponds to a conserved quantity. For example, time-translation symmetry yields energy conservation and spatial translation yields momentum conservation.

A measure of an object's resistance to angular acceleration about a chosen axis, equal to the sum of mass elements times their squared distances from that axis. It depends on both mass distribution and axis.

The rotational effect of a force, equal to the cross product of the position vector from the pivot with the force. It equals the rate of change of angular momentum.

A vector quantity equal to the cross product of position and linear momentum for a particle, summed over a system. It is conserved when net external torque vanishes.

A condition in which a driven oscillator responds with large amplitude because the driving frequency is close to a natural frequency. Damping limits the peak amplitude and shifts it slightly.

A statement that the electric flux through any closed surface equals the enclosed charge divided by the permittivity. It is especially useful for highly symmetric charge distributions.

The scalar electric potential energy per unit charge at a point, defined relative to a reference point. Its negative gradient gives the electric field.

The ratio of stored charge to potential difference across a pair of conductors. For a parallel-plate device it grows with plate area and dielectric constant and shrinks with separation.

An insulating material whose bound charges polarize in an applied field, reducing the internal field for a given free charge. This raises the capacitance of a filled capacitor by the material's relative permittivity.

A relation stating that the line integral of magnetic field around a closed loop equals the enclosed current times the permeability, plus a displacement-current term in time-varying cases. It links currents to magnetic fields.

The surface integral of the magnetic field over a chosen area. A changing value through a conducting loop induces an electromotive force according to Faraday's law.

The proportionality between magnetic flux linkage in a circuit and the current producing it. It opposes changes in current by generating a back electromotive force.

The total force on a charged particle, equal to the charge times the sum of the electric field and the cross product of velocity with the magnetic field. It governs charged-particle motion in fields.

A complex-valued function whose squared modulus gives the probability density of finding a particle at a position and time. It evolves deterministically according to the Schrodinger equation.

The fundamental equation governing how a quantum state evolves in time. For a single particle it relates the rate of change of the wave function to its kinetic and potential energy terms.

The prescription that the probability of a measurement outcome equals the squared modulus of the state's overlap with the corresponding eigenstate. It connects abstract amplitudes to observed frequencies.

A measurable physical quantity represented by a Hermitian operator whose eigenvalues are the possible measurement results. Distinct outcomes correspond to orthogonal eigenstates.

An operator equal to its own adjoint, guaranteeing real eigenvalues and a complete set of orthogonal eigenfunctions. Physical observables are represented by such operators.

The operator combination AB minus BA, measuring the extent to which two operations fail to commute. A nonzero value for two observables implies they cannot be sharply measured together.

A relation, following from noncommuting observables, that limits how precisely two quantities such as position and momentum can be known simultaneously. The product of their uncertainties has a lower bound.

The principle that a quantum state may be a linear combination of other states, with the coefficients setting amplitudes for each. Measurement outcomes are then probabilistic.

A state function measuring the number of microscopic arrangements consistent with a macroscopic state, or equivalently the heat exchanged reversibly divided by temperature. It never decreases in an isolated system.

A principle stating that the total entropy of an isolated system cannot decrease, giving time a preferred direction. It also forbids converting heat entirely into work in a cyclic process.

The maximum possible efficiency of a heat engine operating between two reservoirs, equal to one minus the ratio of cold to hot absolute temperatures. No real engine between the same reservoirs can exceed it.

A sum over all states of the Boltzmann factor, normalizing the canonical ensemble. Most thermodynamic quantities, including free energy and average energy, can be derived from it by differentiation.

A thermodynamic potential that balances internal energy against entropy at fixed temperature, such as Helmholtz or Gibbs energy. A system at fixed conditions tends toward its minimum.

The change in a system's free energy when one particle is added at fixed temperature and volume. It governs diffusion and phase equilibrium between regions or phases.

An integral operation converting a function of time or space into one of frequency or wavenumber, revealing its sinusoidal content. Its inverse reconstructs the original function.

A mathematical combination of two functions that averages one over shifts of the other. In frequency space it becomes simple multiplication, which is why it describes linear system responses.

A distribution that is zero everywhere except at one point, where it is singular, yet integrates to one. It represents an idealized point source or impulse.

An expansion of a smooth function about a point as a sum of terms proportional to powers of the displacement, with coefficients from derivatives. Truncating it gives useful approximations near that point.