GATE PH syllabus 2026

• Linear vector space: basis, orthogonality and completeness
• matrices
• similarity transformations, diagonalization, eigen values and eigen vectors
• linear differential equations: second order linear differential equations and solutions involving special functions
• complex analysis: Cauchy-Riemann conditions, Cauchy's theorem, singularities, residue theorem and applications
• Laplace transform, Fourier analysis
• elementary ideas about tensors: covariant and contravariant tensors.

• D'Alembert's principle, Euler-Lagrange equation, Hamilton's principle, calculus of variations
• symmetry and conservation laws
• central force motion: Kepler problem and Rutherford scattering
• small oscillations: coupled oscillations and normal modes
• rigid body dynamics: inertia tensor, orthogonal transformations, Euler angles, torque free motion of a symmetric top
• Hamiltonian and Hamilton's equations of motion
• Liouville's theorem
• canonical transformations: action-angle variables, Poisson brackets, Hamilton-Jacobi equation.

Lorentz transformations, relativistic kinematics, mass-energy equivalence.

• Solutions of electrostatic and magnetostatic problems including boundary value problems
• method of images
• separation of variables
• dielectrics and conductors
• magnetic materials
• multipole expansion
• Maxwell's equations
• scalar and vector potentials
• Coulomb and Lorentz gauges
• electromagnetic waves in free space, non-conducting and conducting media
• reflection and transmission at normal and oblique incidences
• polarization of electromagnetic waves
• Poynting vector, Poynting theorem, energy and momentum of electromagnetic waves
• radiation from a moving charge.

• Postulates of quantum mechanics
• uncertainty principle
• Schrodinger equation
• Dirac Bracket notation, linear vectors and operators in Hilbert space
• one dimensional potentials: step potential, finite rectangular well, tunneling from a potential barrier, particle in a box, harmonic oscillator
• two and three dimensional systems: concept of degeneracy
• hydrogen atom
• angular momentum and spin
• addition of angular momenta
• variational method and WKB approximation, time independent perturbation theory
• elementary scattering theory, Born approximation
• symmetries in quantum mechanical systems.

• Laws of thermodynamics
• macrostates and microstates
• phase space
• ensembles
• partition function, free energy, calculation of thermodynamic quantities
• classical and quantum statistics: degenerate Fermi gas
• black body radiation and Planck's distribution law
• Bose-Einstein condensation
• first and second order phase transitions, phase equilibria, critical point.

• Spectra of one-and many-electron atoms
• spin-orbit interaction: LS and j couplings
• fine and hyperfine structures
• Zeeman and Stark effects
• electric dipole transitions and selection rules
• rotational and vibrational spectra of diatomic molecules
• electronic transitions in diatomic molecules, Franck-Condon principle
• Raman effect
• EPR, NMR, ESR, X-ray spectra
• lasers: Einstein coefficients, population inversion, two and three level systems.

• Elements of crystallography
• diffraction methods for structure determination
• bonding in solids
• lattice vibrations and thermal properties of solids
• free electron theory
• band theory of solids: nearly free electron and tight binding models
• metals, semiconductors and insulators
• conductivity, mobility and effective mass
• Optical properties of solids
• Kramer's-Kronig relation, intra- and inter-band transitions
• dielectric properties of solid
• dielectric function, polarizability, ferroelectricity
• magnetic properties of solids
• dia, para, ferro, antiferro and ferri- magnetism, domains and magnetic anisotropy
• superconductivity: Type-I and Type-II superconductors, Meissner effect, London equation, BCS Theory, flux quantization.

• Electron and hole statistics in intrinsic and extrinsic semiconductors
• metal-semiconductor junctions
• Ohmic and rectifying contacts
• PN diodes, bipolar junction transistors, field effect transistors
• negative and positive feedback circuits
• oscillators, operational amplifiers, active filters
• basics of digital logic circuits, combinational and sequential circuits, flip-flops, timers, counters, registers, A/D and D/A conversion.

• Nuclear radii and charge distributions, nuclear binding energy, electric and magnetic moments
• semi-empirical mass formula
• nuclear models: liquid drop model, nuclear shell model
• nuclear force and two nucleon problem
• alpha decay, beta-decay, electromagnetic transitions in nuclei
• Rutherford scattering, nuclear reactions, conservation laws
• fission and fusion
• particle accelerators and detectors
• elementary particles
• photons, baryons, mesons and leptons
• quark model
• conservation laws, isospin symmetry, charge conjugation, parity and time-reversal invariance.

Basic English grammar: tenses, articles, adjectives, prepositions, conjunctions, verb-noun agreement, and other parts of speech. Basic vocabulary: words, idioms, and phrases in context. Reading and comprehension, Narrative sequence.

Data interpretation: data graphs (bar graphs, pie charts, and other graphs representing data), 2- and 3-dimensional plots, maps, and tables. Numerical computation and estimation: ratios, percentages, powers, exponents and logarithms, permutations and combinations, and series. Mensuration and geometry. Elementary statistics and probability.

Logic: deduction and induction, Analogy, Numerical relations and reasoning.

Transformation of shapes: translation, rotation, scaling, mirroring, assembling, and grouping paper folding, cutting, and patterns in 2 and 3 dimensions.