1.3.3 Heisenberg Picture

1.3.3 Heisenberg Picture#

Prompts

  • In the Schrödinger picture, states evolve and observables are fixed. What happens in the Heisenberg picture? Are both descriptions physically equivalent?

  • Derive the Heisenberg equation of motion: \(\frac{\mathrm{d}}{\mathrm{d}t}\hat{O}(t) = \frac{\mathrm{i}}{\hbar}[\hat{H}, \hat{O}(t)]\). How does it differ from the Schrödinger equation?

  • For a spin in a magnetic field with \(\hat{H} = \frac{\hbar\omega}{2}\hat{\sigma}^z\) (as in §1.3.2), find how \(\hat{\sigma}^x(t)\) evolves. Describe the motion geometrically.

  • What does it mean for an observable to be conserved? How does the commutator \([\hat{A},\hat{H}] = 0\) guarantee conservation?

  • How do continuous symmetries (like rotations) generate conserved quantities? Give an example from the Standard Model.

Lecture Notes#

Overview#

So far, we have described quantum mechanics in the Schrödinger picture, where states evolve in time and observables remain fixed. There is a mathematically equivalent description—the Heisenberg picture—where operators evolve instead of states. This perspective is more natural for discussing conservation laws and reveals deep connections between quantum and classical mechanics.

Why Two Pictures?#

In the Schrödinger picture, the expectation value of an observable at time \(t\) is:

\[\langle \hat{O} \rangle(t) = \langle \psi(t) \vert \hat{O} \vert \psi(t) \rangle\]

where \(\vert \psi(t) \rangle = \hat{U}(t) \vert \psi(0) \rangle\) and \(\hat{U}(t) = \mathrm{e}^{-\mathrm{i}\hat{H}t/\hbar}\), matching §1.3.2.

We can rewrite this as:

\[\langle \hat{O} \rangle(t) = \langle \psi(0) \vert \hat{U}^\dagger(t) \hat{O} \hat{U}(t) \vert \psi(0) \rangle\]

This factorization suggests two interpretations:

  1. Schrödinger: state \(\vert \psi(t) \rangle\) evolves, operator \(\hat{O}\) is constant.

  2. Heisenberg: state \(\vert \psi \rangle\) is constant, operator \(\hat{O}(t) = \hat{U}^\dagger(t) \hat{O} \hat{U}(t)\) evolves.

Both give identical physical predictions—they are two languages for the same physics.

Heisenberg Equation of Motion#

Heisenberg Equation of Motion

In the Heisenberg picture, operators evolve according to:

\[ \boxed{\frac{\mathrm{d}}{\mathrm{d}t}\hat{O}(t) = \frac{\mathrm{i}}{\hbar}[\hat{H}, \hat{O}(t)]} \]

where \([\cdot,\cdot]\) is the commutator, using the same convention as the Schrödinger equation \(\mathrm{i}\hbar\,\partial_t\vert\psi\rangle = \hat{H}\vert\psi\rangle\).

Derivation

Starting from \(\hat{O}(t) = \hat{U}^\dagger(t) \hat{O} \hat{U}(t)\), we differentiate with respect to time:

\[\frac{\mathrm{d}}{\mathrm{d}t}\hat{O}(t) = \frac{\mathrm{d}\hat{U}^\dagger}{\mathrm{d}t} \hat{O} \hat{U} + \hat{U}^\dagger \hat{O} \frac{\mathrm{d}\hat{U}}{\mathrm{d}t}\]

From the Schrödinger equation for the propagator: \(\mathrm{i}\hbar\,\frac{\mathrm{d}\hat{U}}{\mathrm{d}t} = \hat{H} \hat{U}\), so \(\frac{\mathrm{d}\hat{U}}{\mathrm{d}t} = -\frac{\mathrm{i}}{\hbar}\hat{H} \hat{U}\).

Thus: \(\frac{\mathrm{d}\hat{U}^\dagger}{\mathrm{d}t} = \frac{\mathrm{i}}{\hbar}\hat{U}^\dagger \hat{H}\).

Substituting:

\[\frac{\mathrm{d}}{\mathrm{d}t}\hat{O}(t) = \frac{\mathrm{i}}{\hbar}\hat{U}^\dagger \hat{H} \hat{O} \hat{U} - \frac{\mathrm{i}}{\hbar}\hat{U}^\dagger \hat{O} \hat{H} \hat{U}\]
\[= \frac{\mathrm{i}}{\hbar}\hat{U}^\dagger [\hat{H}, \hat{O}] \hat{U} = \frac{\mathrm{i}}{\hbar}[\hat{H}, \hat{O}(t)]\]

Example: Spin Precession#

Example: Spin in a Magnetic Field

Problem. A spin-1/2 particle evolves under \(\hat{H} = \frac{\hbar\omega}{2}\hat{\sigma}^z\) (the same static-field Hamiltonian as in §1.3.2, with Larmor frequency \(\omega\)). Find \(\hat{\sigma}^x(t)\) and \(\hat{\sigma}^y(t)\) in the Heisenberg picture.

Solution.

The commutators are:

\[[\hat{\sigma}^z, \hat{\sigma}^x] = 2\mathrm{i}\hat{\sigma}^y, \quad [\hat{\sigma}^z, \hat{\sigma}^y] = -2\mathrm{i}\hat{\sigma}^x\]

So, using \(\frac{\mathrm{d}}{\mathrm{d}t}\hat{O} = \frac{\mathrm{i}}{\hbar}[\hat{H}, \hat{O}]\):

\[\frac{\mathrm{d}}{\mathrm{d}t}\hat{\sigma}^x = \frac{\mathrm{i}}{\hbar}\left[\frac{\hbar\omega}{2}\hat{\sigma}^z, \hat{\sigma}^x\right] = -\omega\hat{\sigma}^y\]
\[\frac{\mathrm{d}}{\mathrm{d}t}\hat{\sigma}^y = \frac{\mathrm{i}}{\hbar}\left[\frac{\hbar\omega}{2}\hat{\sigma}^z, \hat{\sigma}^y\right] = \omega\hat{\sigma}^x\]

These are coupled oscillator equations. Taking the second derivative of \(\hat{\sigma}^x\):

\[\frac{\mathrm{d}^2}{\mathrm{d}t^2}\hat{\sigma}^x = -\omega^2\hat{\sigma}^x\]

General solution: \(\hat{\sigma}^x(t) = A\cos(\omega t) + B\sin(\omega t)\).

With initial conditions \(\hat{\sigma}^x(0) = \hat{\sigma}^x\) and \(\frac{\mathrm{d}}{\mathrm{d}t}\hat{\sigma}^x(0) = -\omega\hat{\sigma}^y\):

\[\hat{\sigma}^x(t) = \hat{\sigma}^x\cos(\omega t) - \hat{\sigma}^y\sin(\omega t)\]
\[\hat{\sigma}^y(t) = \hat{\sigma}^y\cos(\omega t) + \hat{\sigma}^x\sin(\omega t)\]

Physical interpretation: In the Heisenberg picture, the spin operators precess around the \(z\)-axis at the Larmor frequency \(\omega\)—the same rate as Bloch-vector precession in the Schrödinger picture (§1.3.2).

Conserved Quantities and Symmetry#

An observable \(\hat{A}\) is conserved if its expectation value does not change in time:

\[\frac{\mathrm{d}}{\mathrm{d}t}\langle \hat{A} \rangle = 0\]

From the Heisenberg equation:

\[\frac{\mathrm{d}}{\mathrm{d}t}\langle \hat{A} \rangle = \frac{\mathrm{i}}{\hbar}\langle [\hat{H}, \hat{A}] \rangle\]

So \(\hat{A}\) is conserved if and only if:

Conservation Law

\[[\hat{A}, \hat{H}] = 0\]

An observable is conserved if and only if it commutes with the Hamiltonian.

This statement connects symmetries to conserved quantities. A symmetry is a unitary transformation \(\hat{U}_\alpha(\theta)\) that leaves the Hamiltonian invariant:

\[\hat{U}_\alpha^\dagger(\theta) \hat{H} \hat{U}_\alpha(\theta) = \hat{H}\]

The generator of the symmetry is an operator \(\hat{G}_\alpha\) such that:

\[\hat{U}_\alpha(\theta) = \mathrm{e}^{-\mathrm{i}\hat{G}_\alpha\theta/\hbar}\]

If \(\hat{U}_\alpha^\dagger \hat{H} \hat{U}_\alpha = \hat{H}\), then \([\hat{G}_\alpha, \hat{H}] = 0\), so \(\hat{G}_\alpha\) is conserved. Every continuous symmetry generates a conserved quantity.

Lie Groups and Quantum Mechanics#

The study of symmetries in quantum mechanics naturally involves Lie groups—continuous families of matrices that form a group under multiplication. Two examples are central to fundamental physics:

U(1): Phase Rotations

The simplest symmetry is global phase rotation:

\[\vert \psi \rangle \to \mathrm{e}^{\mathrm{i}\theta}\vert \psi \rangle\]

This is generated by the identity operator (up to a constant), and is related to particle number conservation in field theory. The group U(1) is one-dimensional.

SU(2): Rotations in Spin Space

The Pauli operators generate rotations in a two-dimensional spin space. Any rotation of a spin-1/2 system can be written as:

\[\hat{U}(\boldsymbol{\theta}) = \mathrm{e}^{-\mathrm{i}\boldsymbol{\theta} \cdot \hat{\boldsymbol{\sigma}} / 2}\]

where \(\boldsymbol{\theta} = (\theta_x, \theta_y, \theta_z)\) and \(\hat{\boldsymbol{\sigma}} = (\hat{\sigma}^x, \hat{\sigma}^y, \hat{\sigma}^z)\). This is the group SU(2), which has three generators (the three Pauli operators). Rotations on the Bloch sphere correspond to SU(2) transformations.

Gauge Symmetries in the Standard Model

The symmetry structure of the electroweak and strong interactions is \(\mathrm{U}(1) \times \mathrm{SU}(2) \times \mathrm{SU}(3)\):

  • U(1): Electromagnetic interactions (photons, electron charge conservation)

  • SU(2): Weak interactions (W and Z bosons)

  • SU(3): Strong interactions (gluons, quark color charge)

Each symmetry group constrains possible interactions and provides conserved currents. This framework, beginning with the idea of operator evolution under symmetry, extends from simple qubits to the fundamental laws of nature.

Discussion

Is symmetry more fundamental than dynamics?

In classical mechanics, we write down a Lagrangian and derive the equations of motion. Symmetries are then discovered as properties of that Lagrangian (Noether’s theorem).

In modern physics, the logic is often reversed: we start with the symmetry group (e.g., \(\mathrm{U}(1) \times \mathrm{SU}(2) \times \mathrm{SU}(3)\)) and derive the allowed interactions. The Standard Model is essentially determined by its gauge symmetries plus the particle content.

  • If symmetry determines dynamics, what determines the symmetry? Why \(\mathrm{SU}(3) \times \mathrm{SU}(2) \times \mathrm{U}(1)\) and not some other group?

  • In condensed matter physics, emergent symmetries appear at low energies that are not present in the microscopic Hamiltonian. Does this mean symmetry is more fundamental than the Hamiltonian, or less?

  • The Heisenberg equation \(\frac{\mathrm{d}}{\mathrm{d}t}\hat{O} = \frac{\mathrm{i}}{\hbar}[\hat{H}, \hat{O}]\) says dynamics = commutator with \(\hat{H}\). Conservation = commuting with \(\hat{H}\). Is the Hamiltonian itself just a way of encoding which symmetries are broken?

Summary#

  • The Heisenberg picture is equivalent to the Schrödinger picture; operators evolve, states are fixed.

  • The Heisenberg equation \(\frac{\mathrm{d}}{\mathrm{d}t}\hat{O} = \frac{\mathrm{i}}{\hbar}[\hat{H}, \hat{O}]\) governs operator dynamics (same \(\hbar\) as \(\hat{U}(t) = \mathrm{e}^{-\mathrm{i}\hat{H}t/\hbar}\) in §1.3.2).

  • Conserved quantities satisfy \([\hat{A}, \hat{H}] = 0\)—they commute with the Hamiltonian.

  • Symmetries are unitary transformations leaving \(\hat{H}\) invariant; their generators are conserved.

  • Lie groups (U(1), SU(2), SU(3), …) describe symmetries; this structure underlies the Standard Model.

Homework#

1. (Picture equivalence) Let \(\vert \psi(t) \rangle = \hat{U}(t) \vert \psi(0) \rangle\) be the Schrödinger-picture state, and define the Heisenberg operator \(\hat{A}(t) = \hat{U}^\dagger(t) \hat{A} \hat{U}(t)\). Show that:

\[\langle\psi(t) \vert \hat{A} \vert \psi(t)\rangle = \langle \psi(0) \vert \hat{A}(t) \vert \psi(0) \rangle\]

This proves both pictures yield identical predictions.

2. (Energy conservation) Show that the Hamiltonian \(\hat{H}\) is always conserved in the Heisenberg picture. Use \(\frac{\mathrm{d}\hat{O}}{\mathrm{d}t} = \frac{\mathrm{i}}{\hbar}[\hat{H}, \hat{O}]\) with \(\hat{O} = \hat{H}\).

3. (Commutation relations) For the Hamiltonian \(\hat{H} = \frac{\hbar\omega}{2}\hat{\sigma}^z\), verify that:

\[[\hat{\sigma}^z, \hat{\sigma}^x] = 2\mathrm{i}\hat{\sigma}^y, \quad [\hat{\sigma}^z, \hat{\sigma}^y] = -2\mathrm{i}\hat{\sigma}^x\]

Use these to check the form of \(\hat{\sigma}^x(t)\) given in the worked example.

4. (Harmonic oscillator) Consider a harmonic oscillator \(\hat{H} = \hbar\omega\left(\hat{a}^\dagger\hat{a} + \frac{1}{2}\right)\). Show that:

\[[\hat{a}, \hat{H}] = -\hbar\omega\hat{a}, \quad [\hat{a}^\dagger, \hat{H}] = \hbar\omega\hat{a}^\dagger\]

Solve the Heisenberg equations \(\frac{\mathrm{d}\hat{a}}{\mathrm{d}t} = \frac{\mathrm{i}}{\hbar}[\hat{H}, \hat{a}]\) and \(\frac{\mathrm{d}\hat{a}^\dagger}{\mathrm{d}t} = \frac{\mathrm{i}}{\hbar}[\hat{H}, \hat{a}^\dagger]\) to find \(\hat{a}(t)\) and \(\hat{a}^\dagger(t)\).

5. (Observable conservation) Prove: if \([\hat{A}, \hat{H}] = 0\), then \(\frac{\mathrm{d}}{\mathrm{d}t}\langle \hat{A} \rangle = 0\) in any quantum state. What is the physical interpretation?

6. (SU(2) generators) Show that the Pauli operators \(\hat{\sigma}^x\), \(\hat{\sigma}^y\), \(\hat{\sigma}^z\) generate infinitesimal rotations on the Bloch sphere via:

\[\hat{U}(\theta) = \mathrm{e}^{-\mathrm{i}\theta\hat{\sigma}^z/2}\]

Verify this rotates \(\hat{\sigma}^x\) to \(\hat{\sigma}^x\cos\theta + \hat{\sigma}^y\sin\theta\) (a rotation by angle \(\theta\) around the \(z\)-axis).

7. (Ehrenfest theorem) Show that the expectation value of position and momentum in a particle obey classical-like equations:

\[\frac{\mathrm{d}}{\mathrm{d}t}\langle \hat{p} \rangle = -\langle \nabla V \rangle, \quad \frac{\mathrm{d}}{\mathrm{d}t}\langle \hat{x} \rangle = \frac{\langle \hat{p} \rangle}{m}\]

These are the quantum Ehrenfest equations—why do they look classical?

8. (Two-state system with zero Hamiltonian) A two-level system has \(\hat{H} = 0\) (no Hamiltonian). What can you say about the time evolution of any observable in the Heisenberg picture? What does this tell you about the dynamics?

9. (Symmetry and conservation) If a unitary transformation \(\hat{U}(\theta) = \mathrm{e}^{-\mathrm{i}\hat{G}\theta/\hbar}\) leaves the Hamiltonian invariant (\([\hat{G}, \hat{H}] = 0\)), then \(\hat{G}\) is conserved. Use this principle to explain: (a) why particle number is conserved in a system with U(1) phase symmetry, and (b) how gauge symmetries in the Standard Model lead to conserved currents (electric charge, weak isospin, color charge).