Index
Chapter 3 · Item 3.12
Heisenberg picture: operator evolution
Operators move, states can be kept fixed
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Guided reading

The Heisenberg picture reorganizes time dependence. Instead of evolving the state vector, it keeps the state fixed and lets operators evolve. The physical predictions do not change; only the representation does. This is why the chapter derives the Heisenberg equation by requiring expectation values to be picture-independent.

Same expectation value, different picture

Let \(|\Psi\rangle_H\) denote the Heisenberg-picture state and \(|\Psi(t)\rangle_S\) the Schrodinger-picture state. The two pictures must give the same expectation value:

\[{}_H\langle\Psi|\hat A_H(t)|\Psi\rangle_H={}_S\langle\Psi(t)|\hat A_S|\Psi(t)\rangle_S.\]

Using \(|\Psi(t)\rangle_S=\hat U(t)|\Psi(0)\rangle_S\), the operator transformation is

\[\hat A_H(t)=\hat U^\dagger(t)\hat A_S\hat U(t).\]

The state is fixed, but the observable carries the time dependence.

Derivative of the Heisenberg operator

Differentiating \(\hat A_H(t)=\hat U^\dagger\hat A_S\hat U\) gives three contributions: the derivative of \(\hat U^\dagger\), any explicit derivative of \(\hat A_S\), and the derivative of \(\hat U\). The evolution equation for \(\hat U\) supplies

\[\frac{d\hat U}{dt}=\frac{\hat H}{i\hbar}\hat U, \qquad \frac{d\hat U^\dagger}{dt}=-\frac{1}{i\hbar}\hat U^\dagger\hat H.\]

After substituting and regrouping terms, the commutator appears naturally.

Heisenberg equation of motion

The result is

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

The second term is present when the observable has explicit time dependence, for example because an external magnetic or electric field is changing with time. If there is no explicit time dependence, the commutator with the Hamiltonian generates the operator's evolution.

Why commutators return

In the measurement section, commutators described compatibility. Here they describe dynamics. If \([\hat H,\hat A_H]=0\) and \(\hat A\) has no explicit time dependence, then \(d\hat A_H/dt=0\). The observable is conserved.

\[[\hat H,\hat A]=0 \quad \Rightarrow \quad \hbox{no time evolution of }\hat A\hbox{ if }\partial\hat A/\partial t=0.\]
Equivalence of pictures

The Schrodinger and Heisenberg pictures are equivalent because all measurable predictions are expectation values and probabilities. As long as states and operators are transformed consistently, the apparatus cannot distinguish which picture was used in the calculation.

Schrodinger pictureHeisenberg picture
States evolve in timeStates can be fixed
Operators usually fixedOperators evolve in time
TDSE for \(|\Psi(t)\rangle\)Heisenberg equation for \(\hat A_H(t)\)
Same expectation valuesSame expectation values
Measurement connection

For Stern-Gerlach calculations, the choice of picture changes how one tracks time before the beam reaches the magnet. The final probabilities are unchanged because the inner products and expectation values are invariant under the picture transformation.

The picture is bookkeeping; the experimental predictions are the physics.
Exercise-ready boundary

This page is designed to support short guided exercises on: Fixed state vectors, time-dependent operators and the Heisenberg equation of motion.

  • 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: Fixed state vectors, time-dependent operators and the Heisenberg equation of motion.
  • Conceptual check: state what the main result says physically before using it algebraically.
  • Equation: \[{}_H\langle\Psi|\hat A_H(t)|\Psi\rangle_H={}_S\langle\Psi(t)|\hat A_S|\Psi(t)\rangle_S.\]
  • Equation: \[\hat A_H(t)=\hat U^\dagger(t)\hat A_S\hat U(t).\]
  • Equation: \[\frac{d\hat U}{dt}=\frac{\hat H}{i\hbar}\hat U, \qquad \frac{d\hat U^\dagger}{dt}=-\frac{1}{i\hbar}\hat U^\dagger\hat H.\]
  • 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.
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Source note: Original auxiliary summary for this book-app, based on Chapter 3 of Mario Reis, Quantum Mechanics, Elsevier, 2026. Book text and figures are copyright © 2026 Elsevier Inc. Cropped figures are shown here with source indication for educational review; consult the original book for the complete presentation.