APS Graduate Exam common mistakes
APS Graduate Exam · Understand the traps and how to avoid them.
Applying a formula in a regime where its assumptions fail (e.g., using the ideal-gas law or equipartition near a phase transition, or non-relativistic kinematics for a particle whose kinetic energy is comparable to its rest energy) without first checking the validity conditions.
Candidates pattern-match on surface features of the problem ('gas', 'fast particle') and reach for the most familiar equation, skipping the order-of-magnitude check that would flag the violated assumption.
What to do instead
Before substituting, write down the assumptions of the chosen relation and verify each numerically (compare kT to interaction energies, v/c to 1, de Broglie wavelength to system size). If an assumption fails, state the correct regime and select the appropriate model.
Dropping or misassigning signs and directions in vector force problems, such as omitting the direction of friction, reversing the relative-velocity sign in drag, or forgetting that tension acts along the string away from the body.
Free-body diagrams are sketched hastily or skipped, and the algebra is done with bare magnitudes so sign errors become invisible until the final answer is nonsensical.
What to do instead
Draw and label a complete free-body diagram with a declared coordinate axis before writing any equation, then keep vector signs explicit through the algebra. Sanity-check the final direction against physical intuition.
Treating a non-conservative or time-dependent situation as if a simple scalar potential existed, for example using energy conservation when friction acts, or writing a potential for a velocity-dependent force.
Energy conservation is over-learned as a universal shortcut, and candidates do not check whether all forces are conservative and whether the potential is time-independent.
What to do instead
List every force and classify it as conservative or not before invoking energy conservation. If non-conservative work is present, use the work-energy theorem with the dissipative term included, or integrate the equation of motion directly.
Confusing field with force and misapplying superposition in electrostatics, e.g., adding potentials as vectors or adding fields as scalars, or forgetting that the field inside a conductor in equilibrium is zero.
Scalar (potential) and vector (field) quantities look similar in notation, and boundary conditions for conductors are memorized as slogans rather than understood.
What to do instead
Label every quantity as scalar or vector at the start. Add potentials algebraically and fields vectorially. For conductors, explicitly invoke E=0 inside and equipotential surfaces before solving.
Getting the boundary conditions or the allowed quantum numbers wrong for a confined system (infinite well, finite well, harmonic oscillator), leading to incorrect energy spectra or normalizable/unnormalizable wavefunctions.
Candidates memorize a single spectrum formula without tracking how continuity of the wavefunction and its derivative, plus normalizability, generate the quantization conditions.
What to do instead
For each problem, write the general solution in each region and impose continuity and normalizability explicitly. Derive the quantization condition rather than recalling it, and verify the ground-state energy is nonzero for confinement.
Misusing operators and commutators: assuming operators commute when they do not, applying an operator to the wrong side, or forgetting that measurement outcomes are eigenvalues and that non-commuting observables cannot share simultaneous eigenstates.
Classical intuition treats multiplication as commutative, and the operator formalism is handled symbolically without tracking ordering or the meaning of expectation values.
What to do instead
Compute commutators explicitly with a test function. Keep operator ordering visible in every line. Before claiming simultaneous definiteness, check that the relevant commutator vanishes.
Confusing state functions with path functions and mishandling exact versus inexact differentials in thermodynamics, e.g., integrating dQ as if it were a state change or assuming entropy is conserved in an irreversible process.
The notation dQ and dW looks identical to exact differentials, and the distinction between reversible and irreversible paths is treated as a formality.
What to do instead
Mark each differential as exact or inexact. For irreversible processes, compute entropy change along a reversible path connecting the same endpoints and verify the second-law inequality. Never assume ΔS=0 unless the process is reversible and adiabatic.
Choosing the wrong statistical ensemble or miscounting microstates, such as applying Maxwell-Boltzmann counting to indistinguishable fermions or bosons, or omitting the degeneracy factor in the partition function.
The counting rules are similar in appearance, and candidates do not check particle distinguishability or occupancy constraints before writing the partition function.
What to do instead
State particle identity (distinguishable, fermion, boson) and constraints first. Build the partition function from the correct single-particle level sum, include degeneracies explicitly, and check limiting behavior at high and low temperature.
Algebraic and calculus slips in multi-step derivations: sign errors in integration by parts, dropping constants of integration, or losing factors in change of variables, which propagate to a wrong final result even when the method is right.
Under time pressure, candidates chain steps mentally and skip intermediate bookkeeping, so a single slip silently corrupts everything downstream.
What to do instead
Write each intermediate step on its own line, carry constants and limits explicitly, and substitute a limiting case or dimensional check at the end to catch errors.
Reporting a measurement without uncertainty or with an incorrectly propagated uncertainty, e.g., adding absolute uncertainties when quantities are multiplied, or ignoring systematic versus statistical error when combining results.
Uncertainty analysis is treated as an afterthought, and the rules for propagation and for combining independent errors are applied from memory without checking the functional form.
What to do instead
Identify the functional relationship first, then propagate using partial derivatives (or quadrature for independent random errors). Separate systematic from statistical contributions, state the confidence level, and report significant figures consistent with the uncertainty.