Dynamics Reference

Use these pages to recover a dynamics formula, translate between quantum pictures, compare Green-function conventions, or find a worked derivation. Each table states the assumptions needed for its formulas. When a boundary condition or approximation matters to your calculation, follow the linked explanation.

Find a dynamics formula

Need	Reference page	What it returns
Define a term quickly	Dynamics Glossary	definition, limitation, and explanation
Recover a core equation	Formula Sheet	convention-fixed formula and route to derivation
Compare pictures, kernels, paths, and phase space	Translation Table of Formulations	equivalent objects and the conditions for each translation
Move states or operators between pictures	Picture Translation Table	dagger, sign, endpoint, and density-operator rules
Look up a standard time-domain kernel	Propagator Table	formula, domain, branch, and boundary reminder
Compare resolvent and response conventions	Green Function Table	$i0$, spectral, density-of-states, and free-kernel identities
Fix a path-integral normalization or source sign	Path Integral Conventions	real-time, Euclidean, measure, source, and Fourier choices
Fix a Wigner or star-product normalization	Phase-Space Conventions	transform, measure, marginal, and bracket conventions
Diagnose a suspicious result	Common Pitfalls	symptom, reason, and correction route
Choose a source or study sequence	Further Reading	annotated source categories and reading routes

Compare the physical problem before the notation

A full-line free-particle kernel does not solve a half-line problem with a reflecting wall. For a Dirichlet wall at $x=0$, the method of images gives, for $x,y>0$,

$$K_D(x,t;y,0)=K_0(x-y,t)-K_0(x+y,t),$$

where $K_0$ is the full-line free kernel. Its even dependence on its spatial argument makes $K_D(0,t;y,0)=0$. The minus sign enforces the boundary condition. For a Neumann wall the image term has a plus sign instead. See Propagators and Boundary Conditions.

Convention differences need a different remedy. Two Wigner formulas may place $2\pi\hbar$ factors differently while describing the same state. Two picture changes may use $R$ and $R^\dagger$ respectively. Translate every related measure, state, and observable consistently before comparing the answers.

Core Structural Checks

Evolution operators

Check

$$U(t_2,t_0) = U(t_2,t_1)U(t_1,t_0),$$ $$U(t,t_0)^\dagger U(t,t_0)=U(t,t_0)U(t,t_0)^\dagger=I,$$

and

$$U(t_0,t_0)=I.$$

For time-dependent Hamiltonians, determine whether time ordering is required. These structural identities are necessary checks, but an evolution generated by the wrong Hamiltonian can satisfy them too.

Propagator kernels

Check the distributional initial condition,

$$K(x,t_0;x',t_0) = \delta(x-x'),$$

the Schrödinger equation at the final endpoint, composition over the correct measure, and the intended boundary conditions.

Green functions

Write the inverse equation and the prescription. Near the spectrum, retain the sign in

$$(E-H\pm i0)^{-1}.$$

Specify whether the object is a time-domain response, an energy-domain resolvent, or a field correlation. Boundary-value signs alone do not identify all of these conventions.

Path integrals

State the regulator, endpoint conditions, measure, real- or imaginary-time weight, source sign, and normalization. A formal $\mathcal Dq$ does not supply these choices by itself.

Phase space

Test normalization and both marginals of a Wigner function. Record the Fourier convention, phase-space measure, and star-product sign before using Moyal dynamics.

References

• P. A. M. Dirac, The Principles of Quantum Mechanics, 4th ed., Oxford University Press, 1958.
• E. N. Economou, Green’s Functions in Quantum Physics, 3rd ed., Springer, 2006.
• M. Hillery, R. F. O’Connell, M. O. Scully, and E. P. Wigner, “Distribution Functions in Physics: Fundamentals,” Physics Reports 106, 121–167 (1984), doi:10.1016/0370-1573(84)90160-1.
• J. J. Sakurai and J. Napolitano, Modern Quantum Mechanics, 3rd ed., Cambridge University Press, 2020.
• L. S. Schulman, Techniques and Applications of Path Integration, Dover, 2005.
• R. Shankar, Principles of Quantum Mechanics, 2nd ed., Springer, 1994.