Dynamics Formulas

Scope

These cards collect the equations most often needed to evolve closed quantum systems, translate between pictures, compose amplitudes, solve source problems, and move between operator and path-integral formulations.

Each card is a calculation reference. It states conventions, assumptions, units, limiting cases, and failure modes, then links to the canonical concept or derivation page. Use the canonical pages when the proof, interpretation, or analytic foundations are the main question.

Choose by task

If you need to…	Start with
Evolve a ket or wavefunction	Schrödinger Equation
Separate a stationary eigenvalue problem	Schrödinger Equation
Propagate many initial states with one operator	Time-Evolution Operator
Expand a noncommuting time-ordered exponential	Dyson Series
Remove a solvable reference evolution	Interaction-Picture Evolution
Evolve observables instead of states	Heisenberg Equation
Evolve expectation values or test classical closure	Ehrenfest Theorem Formulas
Compose coordinate or channel amplitudes	Propagator Composition Law
Invert a differential operator with a prescription	Green Function Equations
Express a fixed-endpoint kernel as a sum over histories	Path-Integral Propagator

Core cards

Schrödinger evolution

Schrödinger Equation collects both forms:

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

and, for a time-independent Hamiltonian after separation of variables,

$$H\lvert n\rangle = E_n\lvert n\rangle.$$

The first is an initial-value equation. The second is a spectral eigenvalue problem. They are related but not interchangeable.

Evolution operators and time ordering

Time-Evolution Operator uses

$$i\hbar\partial_tU(t,t_0) = H(t)U(t,t_0), \qquad U(t_0,t_0)=I.$$

For general time dependence,

$$U(t,t_0) = \mathcal T \exp\!\left[ -\frac{i}{\hbar} \int_{t_0}^{t}H(s)\,ds \right].$$

Dyson Series defines this ordered exponential by nested integrals. Time ordering can be removed only when the generators commute at different times or a special exact reduction has been established.

Pictures of motion

Interaction-Picture Evolution factors

$$U(t,t_0) = U_0(t,t_0)U_I(t,t_0)$$

so that states evolve under a transformed residual interaction. The picture change is exact; truncating the Dyson series for $U_I$ is the approximation.

Heisenberg Equation moves the full closed-system dynamics into operators:

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

Ehrenfest Theorem Formulas takes expectations of the corresponding operator identity. For $H=p^2/(2m)+V(x)$,

$$m\frac{d^2}{dt^2}\langle x\rangle = -\langle V'(x)\rangle.$$

This becomes a closed classical equation for the mean only under additional conditions, exactly for forces affine in position and approximately for some localized states in sufficiently smooth forces.

Kernels and inverse kernels

Propagator Composition Law is the coordinate form of operator multiplication:

$$\begin{aligned} K(q_b,t_b;q_a,t_a) &= \int d\mu(q_c)\, K(q_b,t_b;q_c,t_c)\\ &\qquad\times K(q_c,t_c;q_a,t_a). \end{aligned}$$

Green Function Equations instead concerns an inverse plus a boundary prescription:

$$R(z)=(z-H)^{-1},$$ $$(z-H_q)G(q,q';z) = \delta_\mu(q,q').$$

The retarded time Green function is related to the evolution kernel by

$$G^R(q,t;q',t') = -\frac{i}{\hbar} \theta(t-t')K(q,t;q',t'),$$

under the convention used here. The step function and $-i/\hbar$ factor are part of the definition.

Path-integral representation

Path-Integral Propagator represents the same fixed-endpoint kernel as

$$K(q_f,t_f;q_i,t_i) = \int_{q_i}^{q_f}\mathcal Dq\, e^{iS[q]/\hbar}.$$

The reliable definition begins with finite time slicing. It includes all short-time prefactors, $N-1$ intermediate integrations for $N$ intervals, endpoint data, an ordering rule, and a convergence prescription. The real-time weight is an amplitude, not a probability measure.

Distinctions that prevent errors

Pair	Distinction
Time-dependent vs time-independent Schrödinger equations	Initial-value evolution vs stationary spectral problem.
$U(t,t_0)$ vs $K(q_f,t_f;q_i,t_i)$	Abstract evolution operator vs its coordinate kernel.
$K$ vs $(z-H)^{-1}$	Time propagation of initial data vs energy-domain inverse with a prescription.
$K$ vs $G^R(t,t')$	Evolution kernel vs retarded source kernel, differing by support and normalization.
Heisenberg equation vs Ehrenfest theorem	Operator identity vs its expectation-value consequence.
Interaction picture vs perturbation theory	Exact unitary frame vs a later series truncation.
Dyson series vs ordinary exponential	Ordered products for noncommuting times vs powers of one integrated operator.
Path integral vs classical trajectory	Coherent sum over regulated histories vs one stationary path.

Shared conventions

The formula cards use

$$[A,B]=AB-BA,$$

so

$$\frac{i}{\hbar}[H,A] = \frac{1}{i\hbar}[A,H].$$

Evolution operators act on kets from the right. Thus

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

where the earlier interval appears on the right.

For time-to-energy transforms, the Dynamics convention is

$$F(E) = \int_{-\infty}^{\infty}dt\, e^{iEt/\hbar}F(t),$$

with inverse measure $dE/(2\pi\hbar)$. This convention gives

$$G^R(E)=(E-H+i0)^{-1}$$

for

$$G^R(t) = -\frac{i}{\hbar}\theta(t)e^{-iHt/\hbar}.$$

Changing the Fourier sign changes the displayed $i0$ translation. Always compare defining equations before comparing isolated formulas.

Assumption checklist

Before using a card, identify:

• whether the system is closed and the evolution unitary;
• whether $H$ is time independent, explicitly driven, or split into parts;
• which picture and reference time are in use;
• whether operators are bounded or require common-domain control;
• whether a kernel uses discrete labels, a continuous measure, or both;
• which spatial, temporal, outgoing, incoming, or thermal prescription is imposed;
• whether a series is exact, convergent, asymptotic, or formally truncated;
• whether a path-integral expression is regulated and normalized.

For open systems, use Density Matrices and Open Systems. For transition approximations and scattering, use Approximation and Scattering.

Suggested routes

For a first pass through closed-system dynamics:

1. Schrödinger Equation
2. Time-Evolution Operator
3. Heisenberg Equation
4. Ehrenfest Theorem Formulas
5. Propagator Composition Law

For driven and perturbative dynamics:

1. Interaction-Picture Evolution
2. Dyson Series
3. Time-Dependent Perturbation Theory and Transitions

For alternate formulations and source methods:

1. Propagator Composition Law
2. Path-Integral Propagator
3. Green Function Equations
4. Translation Table of Formulations

Broader references

• Dynamics Formula Sheet
• Propagator Table
• Green Function Table
• Path Integral Conventions
• Pictures of Motion Overview
• Quantum Dynamics

References

• J. J. Sakurai and J. Napolitano, Modern Quantum Mechanics, 3rd ed., Cambridge University Press, 2020.
• R. Shankar, Principles of Quantum Mechanics, 2nd ed., Springer, 1994.
• L. E. Ballentine, Quantum Mechanics: A Modern Development, 2nd ed., World Scientific, 2014.
• R. P. Feynman and A. R. Hibbs, Quantum Mechanics and Path Integrals, McGraw-Hill, 1965.
• E. N. Economou, Green’s Functions in Quantum Physics, 3rd ed., Springer, 2006.