Formula Sheet

This page collects frequently used formulas from quantum dynamics. It is a reference sheet, not a derivation page. For assumptions, domains, and interpretation, follow the links to the canonical discussions.

Schrödinger Equation

For a pure state in the Schrödinger picture,

$$i\hbar\frac{d}{dt}\lvert\psi(t)\rangle = H(t)\lvert\psi(t)\rangle.$$

In a position representation,

$$i\hbar\frac{\partial}{\partial t}\psi(x,t) = \left[ -\frac{\hbar^2}{2m}\frac{\partial^2}{\partial x^2} +V(x,t) \right]\psi(x,t).$$

For a density operator undergoing closed-system unitary dynamics,

$$i\hbar\frac{d\rho}{dt} = [H,\rho].$$

See Liouville–von Neumann Equation for the derivation, invariants, and limits of this closed-system equation.

Evolution Operator Identities

The evolution operator satisfies

$$i\hbar\frac{\partial}{\partial t}U(t,t_0) = H(t)U(t,t_0), \qquad U(t_0,t_0)=I.$$

It composes as

$$U(t_2,t_0)=U(t_2,t_1)U(t_1,t_0),$$

and unitarity gives

$$U^\dagger(t,t_0)U(t,t_0)=I, \qquad U^{-1}(t,t_0)=U(t_0,t).$$

For time-independent $H$,

$$U(t,t_0) = e^{-iH(t-t_0)/\hbar}.$$

If $[H(t),H(t')]=0$ for all relevant times,

$$U(t,t_0) = \exp\left[ -\frac{i}{\hbar}\int_{t_0}^{t}H(t')\,dt' \right].$$

In general,

$$U(t,t_0) = \mathcal T \exp\left[ -\frac{i}{\hbar}\int_{t_0}^{t}H(t')\,dt' \right].$$

Product Formulas

For a time-independent split Hamiltonian $H=H_A+H_B$,

$$e^{-iHt/\hbar} = \lim_{N\to\infty} \left( e^{-iH_A t/(N\hbar)} e^{-iH_B t/(N\hbar)} \right)^N.$$

A common second-order short-time step is

$$e^{-iH\Delta t/\hbar} \approx e^{-iH_A\Delta t/(2\hbar)} e^{-iH_B\Delta t/\hbar} e^{-iH_A\Delta t/(2\hbar)}.$$

See Trotter Product Formula for error scaling, split-operator use, and path-integral connections.

Dyson Expansion

For a time-dependent Hamiltonian,

$$U(t,t_0) = I + \sum_{n=1}^{\infty} \left(-\frac{i}{\hbar}\right)^n \int_{t_0}^{t}dt_1 \int_{t_0}^{t_1}dt_2\cdots \int_{t_0}^{t_{n-1}}dt_n\, H(t_1)\cdots H(t_n).$$

Equivalently, the ordered integration domain may be replaced by explicit time ordering over the full cube.

Picture Transformations

Schrödinger and Heisenberg pictures are related by

$$\lvert\psi_H\rangle=\lvert\psi_S(t_0)\rangle, \qquad A_H(t)=U^\dagger(t,t_0)A_S(t)U(t,t_0).$$

Expectation values agree:

$$\langle\psi_S(t)\rvert A_S(t)\lvert\psi_S(t)\rangle = \langle\psi_H\rvert A_H(t)\lvert\psi_H\rangle.$$

For a split Hamiltonian

$$H(t)=H_0+V(t),$$

the interaction-picture state and interaction operator are commonly defined by

$$\lvert\psi_I(t)\rangle = e^{iH_0(t-t_0)/\hbar}\lvert\psi_S(t)\rangle,$$

and

$$V_I(t) = e^{iH_0(t-t_0)/\hbar}V(t)e^{-iH_0(t-t_0)/\hbar}.$$

Then

$$i\hbar\frac{d}{dt}\lvert\psi_I(t)\rangle = V_I(t)\lvert\psi_I(t)\rangle.$$

Heisenberg Equations

For a Heisenberg operator,

$$\frac{dA_H}{dt} = \frac{i}{\hbar}[H_H,A_H] + \left(\frac{\partial A_S}{\partial t}\right)_H.$$

For canonical variables with

$$[x,p]=i\hbar,$$

and

$$H=\frac{p^2}{2m}+V(x),$$

the equations are

$$\frac{dx_H}{dt}=\frac{p_H}{m}, \qquad \frac{dp_H}{dt}=-V'(x_H).$$

Ehrenfest Relations

For a particle Hamiltonian,

$$\frac{d}{dt}\langle x\rangle = \frac{\langle p\rangle}{m}, \qquad \frac{d}{dt}\langle p\rangle = -\langle V'(x)\rangle.$$

Classical-looking motion follows only under additional conditions, such as narrow wave packets and controlled spreading.

Propagator Kernel

The position-space kernel is

$$K(x_f,t_f;x_i,t_i) = \langle x_f\rvert U(t_f,t_i)\lvert x_i\rangle.$$

It evolves wavefunctions by

$$\psi(x_f,t_f) = \int dx_i\, K(x_f,t_f;x_i,t_i)\psi(x_i,t_i).$$

Its composition law is

$$K(x_f,t_f;x_i,t_i) = \int dx\, K(x_f,t_f;x,t)K(x,t;x_i,t_i).$$

For a one-dimensional free particle,

$$K_0(x_f,t_f;x_i,t_i) = \left[ \frac{m}{2\pi i\hbar (t_f-t_i)} \right]^{1/2} \exp\left[ \frac{im(x_f-x_i)^2}{2\hbar(t_f-t_i)} \right].$$

See Free-Particle Propagator for the momentum-space derivation, composition check, and short-time path-integral role.

For a one-dimensional harmonic oscillator, away from caustic times,

$$\begin{aligned} K_{\rm ho}(x_f,t_f;x_i,t_i) &= \left[ \frac{m\omega}{2\pi i\hbar\sin\omega T} \right]^{1/2} \\ &\quad \times \exp\left\{ \frac{im\omega}{2\hbar\sin\omega T} \left[ (x_f^2+x_i^2)\cos\omega T -2x_fx_i \right] \right\}, \end{aligned}$$

where $T=t_f-t_i$. See Harmonic-Oscillator Propagator for the spectral derivation, classical-action form, and caustic phases.

See Propagator Table for common kernels, boundary-condition warnings, ring and box formulas, and the semiclassical Van Vleck form.

Green Functions and Resolvents

The resolvent is

$$G(z)=(z-H)^{-1}.$$

The retarded and advanced boundary values are often written

$$G^\pm(E) = \lim_{\epsilon\to 0^+} \frac{1}{E-H\pm i\epsilon}.$$

The spectral representation for a discrete nondegenerate spectrum is

$$G(z) = \sum_n \frac{\lvert n\rangle\langle n\rvert}{z-E_n}.$$

See Spectral Representation of Green Functions for discrete and continuous spectral forms, residues, spectral densities, and density-of-states connections.

The density of states can be recovered from the retarded resolvent by

$$\rho(E) = -\frac{1}{\pi} \operatorname{Im} \operatorname{Tr} \frac{1}{E+i0-H}.$$

See Green Functions and Density of States for the derivation and local-density version.

See Retarded and Advanced Green Functions for the time-domain support conditions and the Fourier-transform origin of the $i0$ prescription.

Green functions and time-evolution kernels are related, but they answer different questions: resolvents are energy-domain objects, while propagator kernels are time-domain objects.

See Green Function Table for resolvent, retarded/advanced, spectral-function, density-of-states, and free-particle Green-function conventions.

See What Is a Green Function? for the conceptual distinction between inverse kernels, response functions, spectral resolvents, and propagator kernels.

See Resolvent Operator for the operator definition, spectral-theorem form, and pole interpretation.

Path Integral Formulas

For

$$H=\frac{p^2}{2m}+V(x),$$

the formal real-time expression is

$$K(x_f,t_f;x_i,t_i) = \int_{x(t_i)=x_i}^{x(t_f)=x_f} \mathcal D x(t)\, e^{iS[x]/\hbar},$$

with

$$S[x]=\int_{t_i}^{t_f}dt\, \left[ \frac{m}{2}\dot x^2-V(x) \right].$$

The Euclidean weight is schematically

$$e^{-S_E[x]/\hbar},$$

not an oscillatory phase. Normalization, boundary conditions, and measure definitions depend on the problem.

See Euclidean and Imaginary-Time Path Integrals for Wick rotation, ground-state projection, thermal traces, and Euclidean boundary conditions.

See Time Slicing for the finite-partition construction behind $\mathcal D x(t)$ and the short-time normalization factors.

See Free-Particle Path Integral for the first exact Gaussian path-integral calculation.

See Harmonic-Oscillator Path Integral for the exact oscillator path integral from the classical path and fluctuation determinant.

See Sources and Generating Functionals in QM for source terms, functional derivatives, and correlation-function generation.

See Correlation Functions in Path Integrals for time-ordered, connected, Euclidean, and response correlator conventions.

See Path Integral Conventions for the real-time weight, Euclidean weight, time-slicing normalization, source signs, and Fourier conventions used here.

See Path Integrals from QM to QFT for the translation from coordinate histories to field configurations, source derivatives, and correlation functions.

Wigner–Moyal Formulas

For one degree of freedom, a common Weyl-symbol convention is

$$A_W(x,p) = \int dy\, e^{-ipy/\hbar} \left\langle x+\frac{y}{2} \middle| A \middle| x-\frac{y}{2} \right\rangle.$$

See Weyl Transform for the operator-to-symbol map, Weyl ordering, trace formulas, and the relation to the Wigner function.

For one degree of freedom, a common Wigner transform convention is

$$W(x,p) = \frac{1}{2\pi\hbar} \int dy\, e^{-ipy/\hbar} \left\langle x+\frac{y}{2}\middle|\rho\middle|x-\frac{y}{2}\right\rangle.$$

The marginals are

$$\int dp\,W(x,p)=\langle x\rvert\rho\lvert x\rangle, \qquad \int dx\,W(x,p)=\langle p\rvert\rho\lvert p\rangle.$$

The Wigner function is a quasiprobability distribution, not an ordinary probability density.

See Wigner Function for the density-operator definition, marginals, negativity, Gaussian examples, and the classical-limit bridge.

For a one-mode Gaussian Wigner function with covariance matrix $V$ and mean $\mathbf d$,

$$W(\mathbf z) = \frac{1}{2\pi\sqrt{\det V}} \exp\left[ -\frac12 (\mathbf z-\mathbf d)^T V^{-1} (\mathbf z-\mathbf d) \right].$$

See Gaussian States and Wigner Functions for covariance matrices, coherent states, squeezed states, and quadratic evolution.

See Coherent States in Phase Space for displaced Gaussian Wigner functions, oscillator phase-space trajectories, and coherent-state path-integral previews.

The Moyal bracket is the Wigner–Weyl representation of the commutator:

$$\{A,B\}_M = \frac{1}{i\hbar}(A\star B-B\star A),$$

with

$$A\star B = A\exp\left[ \frac{i\hbar}{2} \left( \overleftarrow{\partial_x}\overrightarrow{\partial_p} - \overleftarrow{\partial_p}\overrightarrow{\partial_x} \right) \right]B.$$

See Star Product for the product rule, $\hbar$ expansion, examples, and trace identities.

See Moyal Bracket for phase-space dynamics, the commutator relation, and the Poisson-bracket limit.

See Phase-Space Dynamics for Wigner–Moyal evolution, quadratic flows, anharmonic corrections, and numerical cautions.

See Phase-Space Conventions for the Wigner transform normalization, Fourier sign, phase-space measure, star-product convention, and Poisson-bracket sign used here.

Algebraic Formulation Formulas

An algebraic state is a positive normalized linear functional

$$\omega:\mathcal A\to\mathbb C, \qquad \omega(I)=1, \qquad \omega(A^*A)\geq0.$$

In a Hilbert-space representation,

$$\omega_\rho(A)=\operatorname{Tr}(\rho A), \qquad \omega_\psi(A)=\langle\psi\vert A\vert\psi\rangle.$$

Heisenberg time evolution is an automorphism of the observable algebra:

$$\alpha_t(A)=U^\dagger(t)AU(t), \qquad \alpha_t(AB)=\alpha_t(A)\alpha_t(B).$$

For Hamiltonian evolution,

$$\frac{d}{dt}\alpha_t(A) = \frac{i}{\hbar}\alpha_t([H,A]).$$

See Algebraic Formulation Overview for observables as an algebra, states as expectation-value functionals, dynamics as automorphisms, and the QFT motivation.

Geometric Quantum Mechanics Formulas

Pure states are rays in projective Hilbert space:

$$\mathbb P(\mathcal H) = (\mathcal H\setminus\{0\})/\mathbb C^\times = S(\mathcal H)/U(1).$$

A self-adjoint operator defines an expectation-value function on rays:

$$f_A([\psi]) = \frac{\langle\psi|A|\psi\rangle} {\langle\psi|\psi\rangle}.$$

For normalized representatives, one common Fubini–Study distance convention is

$$d_{\rm FS}([\phi],[\psi]) = \arccos|\langle\phi|\psi\rangle|.$$

The commutator becomes the projective-space Poisson bracket

$$\{f_A,f_B\}_{\rm FS} = \frac{1}{i\hbar}f_{[A,B]}.$$

Schrödinger evolution projects to Hamiltonian flow generated by $h([\psi])=f_H([\psi])$:

$$\iota_{X_h}\omega_{\rm FS} = dh.$$

See Projective Hilbert Space for rays, projectors, tangent directions, and phase-free pure-state dynamics. See Fubini–Study Geometry for the metric, geodesic distance, qubit example, and quantum-speed relation. See Geometric Quantum Mechanics Overview for Schrödinger flow, Berry holonomy, and the classical-mechanics analogy.

Floquet Formulas

For a periodic Hamiltonian,

$$H(t+T)=H(t),$$

the one-period evolution operator is

$$U(T,0).$$

Floquet states are eigenvectors of this unitary:

$$U(T,0)\lvert u_\alpha(0)\rangle = e^{-i\varepsilon_\alpha T/\hbar} \lvert u_\alpha(0)\rangle.$$

The quasienergy $\varepsilon_\alpha$ is defined modulo $\hbar\Omega$, where $\Omega=2\pi/T$.

See Floquet Theorem in Quantum Mechanics for Floquet states, quasienergies, micromotion, effective Hamiltonians, and the one-period unitary.

See Floquet Operators for the one-period unitary, reference-phase dependence, logarithm branches, stroboscopic observables, and numerical construction.

See Quasienergies for modular spectra, Floquet zones, branch cuts, resonances, and avoided crossings.

Classical-Limit Bridge Formulas

A semiclassical small parameter is dimensionless:

$$\epsilon = \frac{\hbar}{S_0}, \qquad \epsilon\ll1.$$

In suitable semiclassical regimes,

$$\frac{1}{i\hbar}[\hat A,\hat B] \longrightarrow \{A,B\}_{\mathrm{PB}}.$$

The Moyal bracket has the expansion

$$\{A,B\}_M = \{A,B\}_{\mathrm{PB}} + O(\hbar^2).$$

Ehrenfest dynamics gives

$$\frac{d}{dt}\langle x\rangle = \frac{\langle p\rangle}{m}, \qquad \frac{d}{dt}\langle p\rangle = -\langle V'(x)\rangle.$$

Classical Newtonian motion for the packet center additionally requires

$$\langle V'(x)\rangle \approx V'(\langle x\rangle).$$

For an isolated nondegenerate stationary point of

$$I(\hbar) = \int dx\,a(x)e^{iS(x)/\hbar},$$

the leading stationary-phase term is

$$I(\hbar) \sim a(x_\star) e^{iS(x_\star)/\hbar} e^{i\frac{\pi}{4}\operatorname{sgn}S''(x_\star)} \sqrt{ \frac{2\pi\hbar} {\lvert S''(x_\star)\rvert} }.$$

The leading semiclassical propagator has the schematic Van Vleck form

$$K_{\rm sc} = \sum_\gamma \left( \frac{1}{2\pi i\hbar} \right)^{d/2} \lvert D_\gamma\rvert^{1/2} \exp\left[ \frac{i}{\hbar}S_\gamma - i\frac{\pi}{2}\nu_\gamma \right].$$

See What Is the Classical Limit? for the roles of action scaling, stationary phase, wave packets, phase-space flow, decoherence, coarse graining, and semiclassical approximations.

See Stationary Phase for one-dimensional and multidimensional formulas, Hessian phases, endpoint caveats, and path-integral applications.

See Semiclassical Propagator Preview for the action phase, Van Vleck determinant, Maslov phase, caustics, and handoff to detailed semiclassical methods.

Cross-Links

• Time-Dependent Schrödinger Equation
• Time-Evolution Operator
• Time Ordering
• Translation Table of Formulations
• Heisenberg Equations of Motion
• Ehrenfest Theorem
• Propagator Kernel
• Green Function Table
• Path Integral Conventions
• Phase-Space Conventions
• Further Reading
• Common Pitfalls
• Schrödinger Equation Formula Card
• Commutators

References

• J. J. Sakurai and J. Napolitano, Modern Quantum Mechanics, 3rd ed., Cambridge University Press, 2020.
• C. Cohen-Tannoudji, B. Diu, and F. Laloë, Quantum Mechanics, Wiley, 1977.
• R. Shankar, Principles of Quantum Mechanics, 2nd ed., Springer, 1994.
• L. E. Ballentine, Quantum Mechanics: A Modern Development, 2nd ed., World Scientific, 2014.
• L. S. Schulman, Techniques and Applications of Path Integration, Wiley, 1981.