Translation Table of Formulations

Use these tables to translate a specified closed-system calculation. The initial time is $t_0$; $U(t,t_0)$ is the full propagator; $U_0(t,t_0)$ is the propagator for a chosen reference Hamiltonian $H_0(t)$. Assume the propagators exist and the operator products and expectation values below are defined.

Helpful background. Picture Transformations derives the identities.

States and observables in three pictures

Picture	State vector	Observable
Schrödinger	$\lvert\psi_S(t)\rangle=U\lvert\psi_0\rangle$	$A_S(t)$
Heisenberg	$\lvert\psi_H\rangle=\lvert\psi_0\rangle$	$A_H(t)=U^\dagger A_S(t)U$
Interaction	$\lvert\psi_I(t)\rangle=U_0^\dagger\lvert\psi_S(t)\rangle$	$A_I(t)=U_0^\dagger A_S(t)U_0$

The Heisenberg state is fixed even when $H(t)$ varies in time. Explicit time dependence of the measured quantity is retained in $A_S(t)$ in every picture. The same transformation rules apply to density operators by conjugation.

For $H(t)=H_0(t)+V(t)$, define $V_I(t)=U_0^\dagger V(t)U_0$ and $H_H(t)=U^\dagger H(t)U$. Then

$$\begin{aligned} i\hbar\,\partial_t|\psi_S\rangle&=H(t)|\psi_S\rangle,\\ \frac{dA_H}{dt}&=\frac{i}{\hbar}[H_H(t),A_H(t)] +U^\dagger(\partial_t A_S)U,\\ i\hbar\,\partial_t|\psi_I\rangle&=V_I(t)|\psi_I\rangle. \end{aligned}$$

For a time-independent $H$, $H_H=H$. For a general time-dependent Hamiltonian, replacing $H_H(t)$ by the Schrödinger-picture matrix $H(t)$ is not justified. All three pictures give the same scalar expectation value:

$$\langle\psi_S|A_S|\psi_S\rangle =\langle\psi_0|A_H|\psi_0\rangle =\langle\psi_I|A_I|\psi_I\rangle.$$

Evolution operators, kernels, and paths

Object or identity	Translation	Qualification
$U(t_f,t_i)$	$K(x_f,t_f;x_i,t_i)=\langle x_f\rvert U\lvert x_i\rangle$	A position kernel is an amplitude density; use it with a normalized initial wavefunction.
$U(t_2,t_0)=U(t_2,t_1)U(t_1,t_0)$	Integrate the product of kernels over the intermediate position.	Use the configuration-space measure and boundary conditions of the problem.
Spectral expansion of $U$	Sum $e^{-iE_nt/\hbar}\psi_n(x_f)\psi_n^*(x_i)$ for fixed $H$.	A complete discrete basis is assumed; add spectral integrals when needed.
Time-sliced kernel	A regulated path integral with phase $e^{iS/\hbar}$.	The measure and ordering prescription are part of its definition.
Stationary phase	Classical paths plus fluctuations.	Generally an approximation, not a synonym for the spectral expansion.

See Propagator Kernel, Composition Law, and Time Slicing for derivations.

Phase-space translation

Choose the convention in which $\int dq\,dp\,W(q,p)=1$ and the Weyl symbol of the identity is $1$.

Hilbert-space expression	Phase-space expression	What to check
Density operator $\rho$	Wigner function $W$	$W$ can be negative; it is not a joint probability density.
Operator $A$	Weyl symbol $A_W$	Noncommuting products depend on operator ordering.
Product $AB$	$A_W\star B_W$	Ordinary multiplication generally loses quantum corrections.
Commutator divided by $i\hbar$	Moyal bracket	The leading smooth classical expansion is the Poisson bracket.
$\operatorname{Tr}(\rho A)$	$\int dq\,dp\,W A_W$	No extra phase-space factor appears with the stated normalization.

For a Hamiltonian at most quadratic in $q,p$, its Moyal bracket with a sufficiently regular function equals the Poisson bracket exactly. That special simplification does not remove uncertainty constraints on the state. See Phase-Space Conventions and Moyal Bracket. Moyal’s original formulation is listed below.

The field-theory connection

Time ordering, sources, and path integration also organize field theory, but a field propagator is usually a field correlation function, not the position kernel for a single particle. The degrees of freedom, causal interpretation, and need for regularization change. Propagators from QM to QFT makes this comparison explicitly.

Exercises

Suppose $A_S(t)=f(t)I$. What is $A_H(t)$, and which term in its evolution equation produces its derivative?

Solution

Unitarity gives $A_H(t)=f(t)I$. Its commutator with $H_H$ vanishes, but $U^\dagger(\partial_tA_S)U=\dot f(t)I$. Omitting the explicit derivative would incorrectly predict a constant operator. A fixed Heisenberg state does not mean that every observable is constant.

References

• C. Cohen-Tannoudji, B. Diu, and F. Laloë, Quantum Mechanics, Wiley, 1977 — picture transformations with explicit time dependence.
• J. E. Moyal, “Quantum mechanics as a statistical theory,” Mathematical Proceedings of the Cambridge Philosophical Society 45, 99–124, 1949, doi:10.1017/S0305004100000487 — phase-space formulation.
• J. J. Sakurai and J. Napolitano, Modern Quantum Mechanics, 3rd ed., Cambridge University Press, 2020 — pictures and propagators.
• L. S. Schulman, Techniques and Applications of Path Integration, Wiley, 1981 — time-sliced and semiclassical kernels.