Physics GRE

Flashcards to study for the 2026 Physics GRE

period for a simple pendulum

$$T = 2\pi\sqrt{\frac{L}{g}}$$

relativistic gamma

$$\gamma = \frac{1}{\sqrt{1 - (v/c)²}}$$

Relativistic energy

$$E² = (pc)² + (mc²)² = \gamma mc²$$

Equivalent resistance for two resistors in series

$$R_{eq} = R_1 + R_2$$

Equivalent resistance for two resistors in parallel

$$\frac{1}{R_{eq}}=\frac{1}{R_1}+\frac{1}{R_2}$$

Ohm’s law

$$V = IR$$

Photoelectric effect relation

$$K_{max} = h\nu - \phi$$

ground state energy of a particle confined in an infinite square well of length $a$

$$\frac{h²}{8ma²}$$

velocity of light within a nonmagnetic dielectric material

$$v = \frac{c}{\sqrt{\epsilon}}$$

orbital period of a planet around a star

$T = \sqrt{ \frac{4\pi²r³}{GM}}$ , $r$ is the radius of the orbiting body to the star, $M$ is the mass of the central body

velocity of an orbiting body

$$v_{orbit} = \sqrt{ \frac{GM}{r}}$$

angular momentum

$L = I\omega$ or $L = \vec{r} \times \vec{p}$

relationship between angular velocity and linear velocity

$$v = \omega R$$

cyclotron frequency

$$\omega = \frac{qB}{m}$$

quantum efficiency

$$QE = \frac{\text{number of photons detected}}{\text{number of incident photons}}$$

error propagation for $z=x^{n}$ , where $x$ is the measured value

$\frac{\Delta z }{z} = |n| \frac{\Delta x}{x}$ , i.e. multiply the relative uncertainty by $n$

Hooke’s law

$$F = -k\Delta x$$

energy stored in a displaced spring

$$U = \frac{1}{2} k x²$$

resolving power of grating spectrometer

$$P = \frac{ \lambda}{\Delta \lambda}$$

What equation would you use to calculate the index of refraction for a gas in a Michelson interferometer?

$$2d(n-1) = m \lambda$$

$d$ = path length

$n$ = index of refraction

$m$ = number of fringes counted

$\lambda$ = laser wavelength

What are the Pauli spin matrices?

$\sigma_x = \begin{pmatrix} 0 & 1 \\ 1 & 0\end{pmatrix}$ , $\sigma_y = \begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix}$ , $\sigma_z = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}$

binding energy per nucleon

$\text{binding energy per nucleon} = \frac{\Delta m \cdot c²}{A}$ ,

$$\Delta m \text{(mass defect)} = (Z \cdot m_p + N \cdot m_n ) - m_{nucleus}$$

$A$ = mass number

What are fermions?

Half-integer spin (1/2, 3/2, etc.) particles including leptons and quarks.

What are leptons?

The class of particles (specifically fermions) which includes the electron ($e^-$), negative muon ($\mu^-$), and tau ($\tau^-$) particles and their respective neutrinos ($\nu_e,\nu_\mu,\nu_\tau$).

What are hadrons?

Hadrons are the composite subatomic particles made of quarks and antiquarks. The subclasses of hadrons are the mesons and baryons.

What are mesons?

Short-lived particles made from a quark-antiquark pair.

What are baryons?

Composite subatomic particles composed of three quarks. Protons and neutrons are both baryons.

What are bosons?

Integer spin subatomic particles that typically carry the fundamental forces. They do not obey the Pauli exclusion principle.

Gauss’s law for electricity

$$\nabla \cdot E = \frac{\rho}{\epsilon_0}$$

Gauss’s law for magnetism

$$\nabla \cdot B = 0$$

Faraday’s law of induction

$$\nabla \times E = -\frac{\partial B}{\partial t}$$

Ampere-Maxwell’s law

$$\nabla \times B = \mu_0 J + \mu_0 \frac{\partial E}{\partial t}$$

$[\hat{x}, \hat{p}]$ = ?. What does it mean for the commutator of two operators to be zero?

$$[\hat{x}, \hat{p}] = i \hbar$$

If $[ \hat{A}, \hat{B} ] = 0$, then the observables $A$ and $B$ can be measured simultaneously without uncertainty.

Heisenberg’s uncertainty relation (What is the uncertainty bound of $\sigma_x\sigma_p$?)

$$\sigma_x\sigma_p \ge \frac{\hbar}{2}$$

What is the general (time-dependent) Schrödinger equation?

$$E \psi(x,t) = \hat{H} \psi(x,t)$$

$$i\hbar\frac{\partial}{\partial t}\psi(x,t)=\Big(\frac{\hat{p}^{2}}{2m}+\hat{V}(x)\Big)\psi(x,t)$$

What are the position and momentum operators?

$$\hat{x} = x, \hat{p} = -i\hbar \frac{\partial}{\partial x}$$

What is the definition of Hermitian operators and what are three useful facts?

Definition: $\int_{-\infty}^\infty f(x)^* \Big( \hat{A} g(x) \Big) dx = \int_{-\infty}^\infty \Big( \hat{A} f(x) \Big) ^* g(x) dx$

1.) All the eigenvalues of $\hat{A}$ are real.

2.) Eigenfunctions corresponding to different eigenvalues are orthogonal:

$$\int_{-\infty}^\infty f(x)^* g(x) dx = 0$$

3.) $\hat{A} = \hat{A}^\dagger$

$$[AB, C] = \, ?$$

$$[AB, C] = A[B,C] + [A,C]B$$

$$[A, BC] = \, ?$$

$$[A, BC] = [A,B]C + B[A,C]$$

What are the three laws of Thermodynamics?

1.) Energy cannot be created or destroyed. $\Delta U = Q - U$

2.) There is no process in which the sole effect is to transfer heat from a body at a lower temperature to a body at a higher temperature. $\Delta S \ge \int \frac{\delta Q}{T}$ . The entropy of a thermally isolated system cannot decrease.

3.) Entropy is 0 at absolute zero temperature.

What is the ideal gas law and what are the assumptions that are made? What is the van der Waals correction?

$PV = N k_B T$ . Gas molecules of zero size which do not interact with each other. van der Waals: $( P + \frac{N² a }{V²} ) ( V - Nb ) = N k_b T$

What is a reversible thermodynamic process? What are some examples? How do work and entropy change in a reversible process?

Reversible thermodynamic processes proceed in “infinitesimal” steps such that the system is in equilibrium in each step and can be reversed by changing the state of the system (ex. heating a gas within a container of flexible size).

$\delta W = P dV$ (for gasses)

The total entropy change of the system and its surroundings must be zero. The entropy of the system can change by itself:

$$\delta Q = T dS$$

$\Delta S = \int \frac{\delta Q }{T}$ (reversible)

quasistatic process

A process that happens infinitely slowly so that at each instant the system is in thermodynamic equilibrium. (Reversible processes are a subset of quasistatic processes)

adiabatic process

A process for which $\delta Q = 0$ ; no heat is exchanged between the system and its surroundings.

What is an isentropic process? What is the entropy change for such processes? What is the gas equation specific to isentropic processes?

A process that is both adiabatic and reversible (ex. compressing a gas with a piston). The entropy change is 0 for the system and therefore the entropy change for the surroundings is also zero. For a gas undergoing an isentropic process,

$$PV^\gamma = constant$$

where $\gamma = \frac{C_p}{C_V}$

iso-”something” process

A process for which some state variable is held constant. (isobaric = constant pressure, isothermal = constant temperature, isochoric = constant volume).

What is free expansion of a gas?

A irreversible process where a gas suddenly expands (like after popping a balloon). The temperature change during free expansion is constant, so

$PV = P’V’$. There is an entropy change even though $\delta Q = 0$ ; the equation $\Delta S \ge \int \frac{\delta Q }{T}$ does not apply to irreversible processes. The gas also does no work.

What is the Fundamental Thermodynamic Identity? What does it apply to?

$$dU = T dS - PdV$$

This applies to all infinitesimal changes of state.

heat capacity

the amount of heat it takes to change the temperature of an object

heat capacity of constant volume

$$C_{V}=(\frac{\partial Q}{\partial T})_{V}=\frac{\partial U} {\partial T }$$

heat capacity of constant pressure

$$C_p = ( \frac{\partial Q }{\partial T} ) _P$$

What is the relation of $C_P$ and $C_V$ for ideal gasses?

$$C_P - C_V = N k_B$$

specific heat capacity

“intrinsic heat capacity” with units of J g$^{-1}$K$^{-1}$ . It is the amount of heat needed to raise the temperature of an object by one degree.

$$Q = m c \Delta T$$

What is entropy and what is Boltzmann’s equation for it?

It is a measure of the number of microstate corresponding to the system’s macrostate.

$$S = k_B \ln \Omega$$

What is the partition function, $Z$? Why is it useful?

$Z = \sum_j e^{-\beta E_j}$ where $\beta = \frac{1}{k_B T}$.

If you know the partition function, you can compute all state variables.

What is the canonical ensemble of thermodynamics?

The canonical ensemble consists of all possible states of a system with

1.) fixed particle number, $N$

2.) fixed volume, $V$

3.) fixed temperature, $T$

What is the form of the entropy of a monoatomic ideal gas?

$$S = N k_B \ln (\frac{V T^{3/2} } { N} ) + \text{some constants}$$

What is the Equipartition Theorem?

Each quadratic term (degree of freedom) in the Hamiltonian for a particle contributes $\frac{1}{2} k_B T$ to the internal energy of the particle.

Stirling’s formula for $\ln(n!)$

$$\ln(n!) = n(\ln(n) - 1) \text{ for large } n$$