Translation Table of Formulations

This table translates the same closed-system quantum dynamics across common formulations. It is a navigation aid, not a proof that every formulation is equally convenient for every problem.

Core Translation

Concept	Schrödinger picture	Heisenberg picture	Interaction picture
State	$\lvert\psi_S(t)\rangle$ evolves	$\lvert\psi_H\rangle$ fixed for time-independent picture choice	$\lvert\psi_I(t)\rangle$ evolves with transformed interaction
Observable	$A_S$ usually fixed unless explicitly time dependent	$A_H(t)=U^\dagger A_SU$	$A_I(t)=U_0^\dagger A_SU_0$
Evolution object	$U(t,t_0)$ acts on states	$U^\dagger A U$ acts on observables	$U_I(t,t_0)$ evolves interaction states
Main equation	$i\hbar\,d\lvert\psi_S\rangle/dt=H\lvert\psi_S\rangle$	$dA_H/dt=(i/\hbar)[H,A_H]$ plus explicit term	$i\hbar\,d\lvert\psi_I\rangle/dt=V_I(t)\lvert\psi_I\rangle$
Expectation value	$\langle\psi_S(t)\rvert A_S\lvert\psi_S(t)\rangle$	$\langle\psi_H\rvert A_H(t)\lvert\psi_H\rangle$	$\langle\psi_I(t)\rvert A_I(t)\lvert\psi_I(t)\rangle$
Best use	state propagation, wavefunctions	equations of motion, symmetries	perturbations, scattering, light-matter coupling

The expectation value must be the same when the pictures are transformed consistently.

Amplitudes and Kernels

Concept	Operator language	Propagator language	Path-integral language
Transition amplitude	$\langle f\rvert U(t_f,t_i)\lvert i\rangle$	$K(x_f,t_f;x_i,t_i)$	sum over histories from $x_i$ to $x_f$
Composition	$U(t_2,t_0)=U(t_2,t_1)U(t_1,t_0)$	integrate over intermediate positions	concatenate paths at intermediate time
Spectral form	insert energy eigenstates	sum over modes or eigenfunctions	stationary paths and fluctuations
Classical bridge	commutators and Ehrenfest theorem	short-time kernels	stationary phase of $e^{iS/\hbar}$

Phase-Space Translation

Hilbert-space object	Phase-space analogue	Warning
Density operator $\rho$	Wigner function $W(q,p)$	$W$ can be negative
Operator $A$	Weyl symbol $A_W(q,p)$	Operator ordering matters
Commutator	Moyal bracket	Reduces to Poisson bracket only in the classical limit
Expectation value	phase-space integral of $A_W W$	Normalization conventions must be fixed

Field-Theory Bridge

Quantum mechanics	Field-theory continuation	What changes
Evolution operator	time-ordered products and S-matrix	particle number need not be fixed
Propagator kernel	field propagator or correlation function	spacetime and causal structure matter
Path integral over paths	functional integral over fields	measure and renormalization become central
Source coupled to coordinate	source coupled to field	generating functionals organize correlations
Harmonic oscillator modes	field modes	infinitely many modes require regulation

Common Translation Mistakes

Moving time dependence from states to operators but forgetting to transform observables.
Equating a propagator kernel with a Green function without specifying boundary conditions.
Translating a path-integral expression back to operators while ignoring ordering.
Treating Wigner functions as ordinary probability densities.
Importing field-theory terminology into single-particle quantum mechanics before the Hilbert space has changed.

Cross-Links

References

R. Shankar, Principles of Quantum Mechanics, 2nd ed., Springer, 1994.
J. J. Sakurai and J. Napolitano, Modern Quantum Mechanics, 3rd ed., Cambridge University Press, 2020.
H. Weyl, The Theory of Groups and Quantum Mechanics, Dover, 1931.
M. E. Peskin and D. V. Schroeder, An Introduction to Quantum Field Theory, Addison-Wesley, 1995.

Exercises

In which two formulations does the same expectation value appear as state motion in one case and operator motion in the other?

Solution

The Schrödinger and Heisenberg pictures. In the Schrödinger picture,

\langle A\rangle =\langle\psi_S(t)\rvert A_S\lvert\psi_S(t)\rangle.

In the Heisenberg picture,

\langle A\rangle =\langle\psi_H\rvert A_H(t)\lvert\psi_H\rangle.

They agree when $A_H(t)=U^\dagger A_SU$ and $\lvert\psi_S(t)\rangle=U\lvert\psi_H\rangle$ .