GATE PH key terms

A pair of first-order partial differential equations relating the real and imaginary parts of a complex function; satisfying them, together with continuity of the partial derivatives, guarantees the function is complex differentiable.

A result in complex analysis stating that the contour integral of a meromorphic function around a closed curve equals 2πi times the sum of the residues of the poles enclosed by the curve.

A square complex matrix equal to its own conjugate transpose. Its eigenvalues are real, eigenvectors belonging to distinct eigenvalues are orthogonal, and it represents a physical observable in quantum mechanics.

An integral transform that decomposes a function of time or space into a continuous spectrum of frequencies or wave numbers, and whose inverse reconstructs the original function from that spectrum.

A function of generalized coordinates, velocities and time, usually equal to kinetic minus potential energy, from which the equations of motion follow by the Euler–Lagrange equations.

The Legendre transform of the Lagrangian with respect to velocities; it equals the total energy when constraints are time-independent and generates time evolution through Hamilton's equations.

A bilinear operation on functions of canonical coordinates and momenta that encodes their dynamical relation; the time derivative of any observable equals its bracket with the Hamiltonian plus explicit time dependence.

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

Independent collective oscillations of a coupled system in which all parts move at the same frequency with fixed relative amplitudes; any small oscillation is a superposition of these modes.

Four coupled equations relating electric and magnetic fields to their sources: Gauss's laws for electricity and magnetism, Faraday's law of induction, and the Ampère–Maxwell law.

The cross product of electric and magnetic fields divided by the permeability, giving the directional energy flux density of an electromagnetic field, measured in watts per square metre.

A change in the scalar and vector potentials that leaves the electric and magnetic fields unchanged; choosing a gauge such as Lorenz or Coulomb simplifies the field equations.

The distance over which the amplitude of an electromagnetic wave penetrating a conducting medium falls by a factor of e; it decreases as frequency, conductivity or permeability increases.

A complex-valued function of position and time whose squared modulus gives the probability density of finding a particle; it evolves according to the Schrödinger equation.

For two operators A and B, the operator AB − BA. Its value determines whether the corresponding observables can be measured simultaneously with arbitrary precision.

The statement that certain pairs of observables, such as position and momentum, cannot both have arbitrarily precise values; the product of their standard deviations is bounded below by ℏ/2.

A method for finding energy shifts and corrected states when unperturbed levels are degenerate, requiring diagonalization of the perturbation matrix within the degenerate subspace.

The statement that the expectation value of a Hamiltonian in any normalized trial state is never below the true ground-state energy, providing an upper bound used to approximate that energy.

A sum over all microstates of the Boltzmann factor exp(−E/kT); from it, thermodynamic quantities such as free energy, entropy and mean energy are obtained by differentiation.

The change in a thermodynamic potential when one particle is added at constant temperature and volume; it governs diffusion and phase equilibrium and appears in quantum distribution functions.

The average occupancy of a quantum state of energy E by identical fermions, equal to 1/[exp((E−μ)/kT)+1]; at zero temperature it is a step function at the Fermi energy.

The accumulation of a macroscopic fraction of identical bosons in the lowest quantum state below a critical temperature, arising because any number of bosons may occupy one state.

Conditions, usually on changes in quantum numbers such as angular momentum and parity, that determine whether a transition between two states is allowed by a given interaction.

Small splittings of atomic energy levels caused by relativistic corrections to kinetic energy, spin–orbit coupling and the Darwin term; observed as close multiplets in spectral lines.

The splitting of atomic spectral lines when an external magnetic field is applied, caused by interaction of the field with the orbital and spin magnetic moments of the electrons.

The set of wave vectors that yield plane waves with the periodicity of a crystal lattice; it is used to describe diffraction conditions and electronic band structure.

A primitive cell of the reciprocal lattice, conventionally the Wigner–Seitz cell; wave vectors within it label distinct Bloch states of a periodic crystal.

For a periodic potential, energy eigenstates have the form of a plane wave multiplied by a function with the period of the lattice, giving band structure and crystal momentum.

A quantized collective vibration of a crystal lattice, characterized by a wave vector and polarization; phonons carry heat and scatter electrons in solids.

The surface in wave-vector space separating occupied from unoccupied electron states at zero temperature; its shape determines transport and thermodynamic properties of metals.

The energy required to separate a bound system, such as a nucleus, into its constituents; for a nucleus it reflects the mass defect through E = mc².

A measure of the probability of a scattering or reaction process, expressed as an effective area; the number of events per unit time equals the incident flux times the cross section.

A scheme describing hadrons as bound states of quarks and antiquarks carrying colour charge; mesons contain a quark–antiquark pair and baryons contain three quarks.

The interface between p-type and n-type semiconductor regions, where diffusion creates a depletion layer and a built-in potential that controls current flow.

A high-gain differential amplifier, usually integrated, whose behaviour is set by external feedback components; ideal versions have infinite gain and infinite input impedance.

Returning a fraction of an amplifier's output to its input with opposite phase, which stabilizes gain, widens bandwidth, reduces distortion and changes input and output impedances.

A physical device implementing a Boolean operation such as AND, OR or NOT on binary inputs; combinations of gates form digital circuits used in computation and control.

A form of deductive reasoning with two premises sharing a middle term, from which a conclusion about the other terms follows if the premises are true.

The aptitude skill of reading tables, charts and graphs, extracting the relevant figures and drawing correct numerical conclusions from them accurately and quickly.