Which Formulation Should I Use?

The best formulation is the one that makes the question simple while keeping the assumptions visible. None is universally superior.

Quick Decision Guide

Task	Start with	Why
Find the state at time $t$	Schrödinger picture	It evolves $\lvert\psi(t)\rangle$ directly
Show a probability is conserved	Unitary evolution	Norm preservation is explicit
Track an observable such as $x(t)$ or $S_z(t)$	Heisenberg picture	Operators obey equations of motion
Treat a weak time-dependent perturbation	Interaction picture	The solvable part is factored out
Compute amplitude from one position to another	Propagator kernel	It gives the coordinate-space transition amplitude
Extract spectral information	Resolvent or Green function	Poles and branch cuts encode spectra
Use the action and stationary phase	Path integral	Histories are weighted by $e^{iS/\hbar}$
Compare with classical phase space	Wigner-Moyal formulation	It represents states and observables on phase space
Study a periodic drive	Floquet formulation	One-period evolution becomes central
Prepare for field theory	Interaction picture, Green functions, path integrals	These become standard field-theory tools

Common Scenarios

I Need the Time-Dependent Wavefunction

Use the Schrödinger picture or an explicit time-evolution operator:

$$\lvert\psi(t)\rangle =U(t,t_0)\lvert\psi(t_0)\rangle.$$

For wave mechanics, propagators and numerical methods can also be useful, but the conceptual home is state evolution.

I Need Expectation Values of Operators

If the operator has simple dynamics, use the Heisenberg picture:

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

This is especially efficient for oscillator ladder operators, spin precession, constants of motion, and symmetry arguments.

I Have a Small Perturbation Switched On in Time

Use the interaction picture. Write

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

solve the $H_0$ motion exactly, and let the interaction-picture state evolve with the transformed perturbation. Detailed transition-rate calculations belong in approximation and scattering pages, but the formal language begins here.

I Need a Transition Amplitude

Use a propagator:

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

The propagator is an amplitude, not a probability. Probabilities require taking appropriate absolute squares or integrating against wavefunctions.

I Need Spectra or Boundary Conditions

Use the resolvent or a Green function. The resolvent

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

packages spectral information and becomes especially important in scattering, response theory, and many-body physics.

I Want the Classical Limit

Use more than one tool. Ehrenfest theorem gives equations for expectation values, stationary phase explains why classical paths dominate certain limits, and the Phase-Space Formulation explains how Wigner functions place quantum states on phase space without turning them into ordinary probability distributions.

What Not to Do

• Do not use $e^{-iH(t-t_0)/\hbar}$ for a time-dependent Hamiltonian unless the relevant Hamiltonians commute at different times or the expression has been justified.
• Do not treat a propagator as a probability.
• Do not use path integrals to bypass operator-ordering questions.
• Do not assume the Heisenberg picture is only philosophical; it is often the most efficient calculational language.
• Do not use the interaction picture as a synonym for perturbation theory. It is an exact rewriting before approximations are made.

Cross-Links

• Translation Table of Formulations
• Unitary Time Evolution
• Time-Dependent Schrödinger Equation
• Phase-Space Formulation
• Why Phase Space in Quantum Mechanics?
• Commutators
• Path Integrals

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.
• C. Cohen-Tannoudji, B. Diu, and F. Laloë, Quantum Mechanics, Wiley, 1977.

Exercises

1. A Hamiltonian is $H(t)=H_0+V(t)$, where $H_0$ is exactly solvable and $V(t)$ is weak. Which formulation should you try first?

Solution

The interaction picture is the natural first choice. It factors out the exactly solvable $H_0$ evolution and leaves the transformed perturbation to drive the remaining dynamics.