Unitary Time Evolution and Quantum Pictures: Schrödinger, Heisenberg, and Time-Ordered Exponentials | MIT OCW

Overview

In this MIT OpenCourseWare lecture, the instructor develops unitary time evolution from a given Hamiltonian and introduces the Schrödinger and Heisenberg pictures of quantum mechanics. The session covers solving for the time evolution operator in time independent and commuting time dependent cases, explains the time ordered exponential for general non commuting Hamiltonians, and discusses the physical interpretation of wavefunction evolution. A harmonic oscillator example illustrates how Heisenberg operators obey classical like equations of motion and how the Heisenberg and Schrödinger pictures yield equivalent predictions for observables.

Key topics

Includes deriving the Schrödinger equation from time evolution, the role of commutators, and the connection to Poisson brackets. The lecture emphasizes that the unitary evolution operator is the central object for dynamics and explains how energy eigenstates transform under time evolution.

Introduction and Plan

This lecture from MIT OpenCourseWare surveys unitary time evolution in quantum mechanics starting from the postulate of time dependent states and a time evolution operator U. The instructor derives the Schrödinger equation from the time derivative of the evolution operator and emphasizes that U depends on the Hamiltonian in a way that is independent of the initial time. The talk also introduces the Heisenberg picture, where operators carry time dependence while states remain fixed at the initial time, establishing the bridge between quantum dynamics and observable quantities.

From Schrödinger to Heisenberg

The core idea is to define Heisenberg operators AHeisenberg as U† ASchroedinger U. This construction ensures that matrix elements computed with time dependent states can be re-expressed as expectation values using time zero states and Heisenberg operators. The lecturer highlights the algebraic rules that carry over between pictures, such as how products and commutators transform, and notes that the identity operator remains the same in both pictures, since U0 is the identity at t = 0.

Time Evolution in Special Cases

The discussion then focuses on three cases for the Hamiltonian H(t). Case 1 considers time independent H, where the evolution operator is simply the exponential of the Hamiltonian, U(t) = exp(-iHt/ħ). Case 2 deals with a time dependent H that commutes at all times with itself, so the exponential form generalizes to a time integral, and the derivative behaves like a standard exponential. Case 3 addresses the general, non commuting time dependent case, where a time-ordered exponential is required. The speaker explains the structure of the time-ordered exponential through a series expansion with nested integrals, and clarifies its role as the formal solution when time dependence does not commute at different times.

Heisenberg Equations of Motion and Observables

A key result is the Heisenberg equation of motion, which gives the time derivative of Heisenberg operators in terms of commutators with the Heisenberg Hamiltonian, plus any explicit time dependence. The lecture derives D AHeisenberg/dt = (i/ħ)[HHeisenberg, AHeisenberg] + (∂A/∂t)Heisenberg. The implications for conserved quantities and expectation values are discussed, along with the fact that expectation values of Schrödinger operators equal those of their Heisenberg counterparts when evaluated with time zero states.

Harmonic Oscillator Example