Quantum Dynamics: Schrödinger, Heisenberg, and Path Integral Formulations

Manual: General · Subject: Quantum Physics

Compare equivalent dynamical formulations and their roles in perturbation theory, semiclassics, and field theory.

Three views of dynamics

Schrödinger picture

In the Schrödinger picture, states evolve in time while operators are typically fixed. Dynamics obey $i\hbar \partial_t |\psi(t)\rangle = \hat{H}|\psi(t)\rangle$ .

Heisenberg picture

In the Heisenberg picture, operators evolve as $\hat{A}(t)=U^\dagger(t)\hat{A}(0)U(t)$ while states remain fixed. This is especially useful for symmetries, correlation functions, and quantum field theory.

Path integral formulation

The transition amplitude can be written as a sum over histories, schematically $\langle x_f,t_f|x_i,t_i\rangle = \int \mathcal{D}x\, e^{\frac{i}{\hbar}S[x]}$ . This formulation is natural for semiclassics, gauge theories, and many-body systems.

ℹ️

Equivalence

For well-posed systems, the Schrödinger, Heisenberg, and path integral formulations are mathematically equivalent ways of encoding the same physics.

Using the path integral in practice

1

Step 1: Write the action $S[x]$ or $S[\phi]$ .
2

Step 2: Identify the classical stationary path $\delta S = 0$ .
3

Step 3: Expand around the saddle to obtain semiclassical corrections.
4

Step 4: Include fluctuations, sources, and symmetry constraints.

What quantity appears in the phase weight of the path integral?

e^{-S/\hbar}

e^{iS/\hbar}

e^{S/\hbar}

e^{-iS\hbar}

Why is the Heisenberg picture especially convenient for correlation functions?

In the semiclassical approximation to the path integral, the dominant contribution comes from which paths?

Paths with maximal kinetic energy Classical stationary-action paths Paths with zero phase Random paths chosen uniformly

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