Index
Chapter 2 · Item 2.11
Equations of motion: Ehrenfest and Virial
How expectation values move
11 / 15
Guided reading

Ehrenfest and Virial results show how wave mechanics talks to classical mechanics. They do not erase quantum behavior; they explain why averages can sometimes obey equations that resemble classical laws.

Time derivative of an expectation value

The Schrodinger equation implies a compact equation for the time evolution of expectation values:

\[\frac{d}{dt}\langle \hat A\rangle=\left\langle\frac{\partial \hat A}{\partial t}\right\rangle+\frac{i}{\hbar}\langle[\hat H,\hat A]\rangle.\]

The first term is explicit time dependence of the operator. The second term comes from the noncommuting relation between the observable and the Hamiltonian.

A direct consequence is important: if \(\hat A\) has no explicit time dependence and \([\hat H,\hat A]=0\), then \(\langle\hat A\rangle\) is constant in time. The commutator therefore tells which average quantities are conserved.

Ehrenfest theorem

For the usual Hamiltonian \(\hat H=\hat p^2/2m+V(\hat x)\), the position and momentum averages obey

\[\frac{d}{dt}\langle \hat x\rangle=\frac{\langle \hat p\rangle}{m},\qquad \frac{d}{dt}\langle \hat p\rangle=-\left\langle V'(\hat x)\right\rangle.\]

This resembles Newton's law for expectation values, but the force is averaged over the quantum state. If the packet is narrow enough that \(\langle V'(\hat x)\rangle\approx V'(\langle\hat x\rangle)\), the mean position follows an approximately classical trajectory.

Virial theorem

For bound stationary states, the virial theorem relates average kinetic and potential behavior:

\[2\langle T\rangle=\left\langle x\frac{dV}{dx}\right\rangle.\]

If \(V(x)\propto x^2\), then \(x\,dV/dx=2V\), so \(\langle T\rangle=\langle V\rangle\). If \(V(r)\propto -1/r\), the relation gives the familiar Coulomb balance.

Conceptual bridge

Ehrenfest's theorem does not restore classical trajectories for individual particles. It shows that certain averages can follow classical-looking equations when the state is sufficiently localized or when the potential is smooth over the spread of the wave packet.

Why expectation values can look classical

Ehrenfest's theorem tracks the time evolution of averages, not individual particle trajectories. For position and momentum it gives the quantum analogue of Newton's equations:

\[\frac{d\langle x\rangle}{dt}=\frac{\langle p\rangle}{m},\qquad \frac{d\langle p\rangle}{dt}=-\left\langle\frac{dV}{dx}\right\rangle.\]

The result becomes classically transparent when the wave packet is narrow enough that the average force is close to the force at the average position.

The Virial theorem is a separate average-energy statement; exercises should not mix the two without stating which average is being studied.

Exercise-ready boundary

This page is designed to support short guided exercises on: Time evolution of expectation values, the classical limit and the Virial theorem.

  • Use from this page: the definitions, physical setup, highlighted equations and conceptual links needed to start a first calculation or explanation.
  • Keep in the book: complete derivations, extended historical discussion, worked solutions and the full textbook narrative remain in the original chapter and linked book resources.
  • Good exercise balance: ask the student to identify assumptions, apply one relation, and interpret the result physically, without requiring material not introduced on this page.
Practice anchors

Use these anchors to design compact exercises. The exercise should be answerable from this page plus standard algebra, while longer derivations, full worked examples and broader context should point back to the original book.

  • Focus: Time evolution of expectation values, the classical limit and the Virial theorem.
  • Conceptual check: state what the main result says physically before using it algebraically.
  • Equation: \[\frac{d}{dt}\langle \hat A\rangle=\left\langle\frac{\partial \hat A}{\partial t}\right\rangle+\frac{i}{\hbar}\langle[\hat H,\hat A]\rangle.\]
  • Equation: \[\frac{d}{dt}\langle \hat x\rangle=\frac{\langle \hat p\rangle}{m},\qquad \frac{d}{dt}\langle \hat p\rangle=-\left\langle V'(\hat x)\right\rangle.\]
  • Equation: \[2\langle T\rangle=\left\langle x\frac{dV}{dx}\right\rangle.\]
  • Boundary: use this page for setup and first-step reasoning; cite the book for longer derivations, complete experimental history or solved-problem detail.
  • Typical task: derive, interpret, or apply the relation above to a simple case without introducing topics outside this page.
Original book and previews:
Elsevier shopScienceDirectGoogle Books previewAmazon
Source note: Original auxiliary summary for this book-app, based on Chapter 2 of Mario Reis, Quantum Mechanics, Elsevier, 2026. Book text and figures are copyright © 2026 Elsevier Inc. No original book figure is reproduced on this page.