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 \(\hat{H} = \frac{\hbar\omega}{2}\hat{\sigma}^z\), 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?
If a Hamiltonian commutes with a symmetry operation (like a rotation or reflection), what does this tell you about the structure of the system and its possible outcomes?

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:

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

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?

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?

Poll: Conservation laws in Heisenberg picture

In the Heisenberg picture, observables satisfy \(\frac{\mathrm{d}}{\mathrm{d}t}\hat{A}_H = \frac{\mathrm{i}}{\hbar}[\hat{H}, \hat{A}_H]\). If \([\hat{H}, \hat{A}] = 0\), what can you conclude?

(A) \(\hat{A}_H(t) = \hat{A}\) (constant, a conserved quantity).

(B) \(\vert\langle\hat{A}_H\rangle\vert\) is constant in time.

(D) The eigenvalues of \(\hat{A}_H\) are time-dependent.

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 on a concrete example. Take the Hamiltonian \(\hat H = \tfrac{\hbar\omega}{2}\hat\sigma^z\) and the initial state \(\vert\psi(0)\rangle = \vert+\rangle\). Compute \(\langle\hat\sigma^x\rangle(t)\) in two ways and verify the two answers agree.

(a) Schrödinger picture. Evolve the state \(\vert\psi(t)\rangle = \hat U(t)\vert+\rangle\) using \(\hat U(t) = \mathrm{e}^{-\mathrm{i}\omega t\hat\sigma^z/2}\) from 1.3.2. Compute \(\langle\psi(t)\vert\hat\sigma^x\vert\psi(t)\rangle\).

(b) Heisenberg picture. Compute the time-evolved operator \(\hat\sigma^x(t) = \hat U^\dagger(t)\hat\sigma^x\hat U(t)\) using the lecture’s example (or the Heisenberg equation of motion). Compute \(\langle+\vert\hat\sigma^x(t)\vert+\rangle\).

(c) Verify the two methods give the same result, and identify exactly which algebraic identity collapses one calculation into the other.

2. Heisenberg evolution under a tilted Hamiltonian. The lecture computes Heisenberg evolution for \(\hat H = \tfrac{\hbar\omega}{2}\hat\sigma^z\). Now take the tilted Hamiltonian

\[ \hat H = \tfrac{\hbar\omega}{2\sqrt 2}\bigl(\hat\sigma^x + \hat\sigma^z\bigr) = \tfrac{\hbar\omega}{2}\,\boldsymbol{n}\cdot\hat{\boldsymbol\sigma}, \]

with \(\boldsymbol{n} = (1, 0, 1)/\sqrt 2\) — a unit vector in the \(xz\)-plane at \(45^\circ\) from \(\boldsymbol{e}_z\).

(a) Compute the Heisenberg-equation rates \(\mathrm{d}\hat\sigma^x/\mathrm{d}t\), \(\mathrm{d}\hat\sigma^y/\mathrm{d}t\), \(\mathrm{d}\hat\sigma^z/\mathrm{d}t\) using \([\hat\sigma^a,\hat\sigma^b] = 2\mathrm{i}\epsilon^{abc}\hat\sigma^c\).

(b) Identify which Pauli operator (if any) is conserved under this Hamiltonian. Express the conserved operator in terms of the original \(\hat\sigma^a\).

(c) Find \(\hat\sigma^y(t)\) explicitly. Hint: the two equations involving \(\hat\sigma^y\) and the conserved combination decouple from the third equation.

3. Pauli precession as a cross product. For the general qubit Hamiltonian \(\hat H = \tfrac{\hbar}{2}\boldsymbol\omega\cdot\hat{\boldsymbol\sigma}\) with \(\boldsymbol\omega \in \mathbb{R}^3\) a constant vector:

(a) Show by direct computation that the Heisenberg equation \(\mathrm{d}\hat\sigma^a/\mathrm{d}t = (\mathrm{i}/\hbar)[\hat H,\hat\sigma^a]\) becomes

\[ \frac{\mathrm{d}\hat\sigma^a}{\mathrm{d}t} = \epsilon^{abc}\,\omega^b\,\hat\sigma^c \quad\equiv\quad \frac{\mathrm{d}\hat{\boldsymbol\sigma}}{\mathrm{d}t} = \boldsymbol\omega\times\hat{\boldsymbol\sigma}. \]

(b) Take expectation values on any state and conclude that the Bloch vector \(\boldsymbol n(t) = \langle\hat{\boldsymbol\sigma}\rangle\) obeys the classical precession equation

\[ \frac{\mathrm{d}\boldsymbol n}{\mathrm{d}t} = \boldsymbol\omega\times\boldsymbol n. \]

(c) Recognize this as the classical precession of a magnetization vector about a magnetic field: \(\boldsymbol\omega\) plays the role of \(\gamma\boldsymbol B\). Interpret the magnitude \(\vert\boldsymbol\omega\vert\) and the direction \(\boldsymbol{\omega}/\omega\) in geometric terms — what does each control about the trajectory of \(\boldsymbol n\)?

★ 4. Harmonic oscillator dynamics. Consider a harmonic oscillator \(\hat{H} = \hbar\omega\left(\hat{a}^\dagger\hat{a} + \frac{1}{2}\right)\), with ladder operators satisfying \([\hat{a}, \hat{a}^\dagger] = 1\). Show that:

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

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. Conservation at the expectation level. A subtle distinction: an observable’s expectation value on a specific state can be time-independent even when the operator itself is not conserved (does not commute with \(\hat H\)).

Take \(\hat H = \tfrac{\hbar\omega}{2}\hat\sigma^z\).

(a) Verify that \(\hat\sigma^x\) is not conserved as an operator: from the lecture’s worked example, \(\hat\sigma^x(t) = \hat\sigma^x\cos(\omega t) - \hat\sigma^y\sin(\omega t) \neq \hat\sigma^x\).

(b) Find a pure qubit state \(\vert\psi\rangle\) for which the expectation value \(\langle\psi\vert\hat\sigma^x(t)\vert\psi\rangle\) is constant in time, despite (a). Identify all such states.

(d) For a single observable \(\hat A\), the operator satisfies \([\hat A,\hat H]=0 \Rightarrow \hat A(t) = \hat A\). State (without proof) when the expectation value of a non-commuting \(\hat A\) is nevertheless conserved on a given state.

6. SU(2) generator and operator rotation. Show that conjugation of \(\hat\sigma^x\) by \(\mathrm{e}^{-\mathrm{i}\theta\hat\sigma^z/2}\) rotates \(\hat\sigma^x\) to \(\hat\sigma^x\cos\theta - \hat\sigma^y\sin\theta\) — the Heisenberg-picture mirror of 1.3.1 Problem 6 (where the state rotated about \(\boldsymbol{e}_z\) instead of the operator).

(a) Define \(\hat\sigma^x(\theta) = \mathrm{e}^{-\mathrm{i}\theta\hat\sigma^z/2}\,\hat\sigma^x\,\mathrm{e}^{+\mathrm{i}\theta\hat\sigma^z/2}\). Differentiate with respect to \(\theta\) and show that \(\mathrm{d}\hat\sigma^x(\theta)/\mathrm{d}\theta = -(\mathrm{i}/2)[\hat\sigma^z,\hat\sigma^x(\theta)]\).

(b) Evaluate at \(\theta = 0\) using \([\hat\sigma^z,\hat\sigma^x] = 2\mathrm{i}\hat\sigma^y\). Identify the infinitesimal generator action \(\hat\sigma^x(\theta)\big\vert_{\theta\to 0} \approx \hat\sigma^x + \theta\hat\sigma^y\).

(c) Iterate the differential equation (or use the closed-form \((\hat\sigma^z)^2 = \hat I\) expansion) to find \(\hat\sigma^x(\theta)\) exactly: \(\hat\sigma^x(\theta) = \hat\sigma^x\cos\theta - \hat\sigma^y\sin\theta\). (Sign of \(\sin\theta\) tracks the conjugation order — Heisenberg-picture time evolution conjugates with \(\hat U^\dagger\) on the left.)

(d) Connect to 1.3.1 Problem 6: conjugation by \(\mathrm{e}^{-\mathrm{i}\theta\hat\sigma^z/2}\) rotates the operator about \(\boldsymbol{e}_z\) by angle \(\theta\); in 1.3.1 Problem 6 the state rotated about \(\boldsymbol{e}_z\) by the same angle. Why are these two equivalent descriptions of the same physics?

7. Cyclic evolution and the half-angle. Take the Hamiltonian \(\hat H = \tfrac{\hbar\omega}{2}\hat\sigma^z\).

(a) Compute \(\hat U(t) = \mathrm{e}^{-\mathrm{i}\omega t\hat\sigma^z/2}\) at \(t = 2\pi/\omega\) and \(t = 4\pi/\omega\). Identify the period \(T_{\hat U}\) after which \(\hat U\) returns to \(+\hat I\).

(b) Compute the Heisenberg-evolved Pauli \(\hat\sigma^x(t)\) from the lecture’s formula and identify the period \(T_{\hat\sigma}\) after which \(\hat\sigma^x(t) = \hat\sigma^x\).

(c) Show that \(T_{\hat U} = 2T_{\hat\sigma}\). Explain in one sentence why the unitary takes twice as long to return to the identity as the Pauli operator does to return to itself.

(d) Apply \(\hat U(2\pi/\omega) = -\hat I\) in the Schrödinger picture and in the Heisenberg picture: what happens to a state under \(\hat U(2\pi/\omega)\), and what happens to an operator under conjugation by \(\hat U(2\pi/\omega)\)? Why is the global \(-1\) visible in the state-vector but invisible in the conjugated operator?

8. Algebra of conserved quantities. A conserved observable \(\hat A\) satisfies \([\hat A,\hat H] = 0\).

(a) Suppose \(\hat A\) and \(\hat B\) are both conserved. Show that \([\hat A + \hat B,\hat H] = 0\) and \([\hat A\hat B,\hat H] = 0\). (Hint: distributivity and product rule for the commutator.)

(b) Use this to argue that the set of all operators commuting with \(\hat H\) is an algebra — closed under addition, scalar multiplication, and operator multiplication.

(i) verifying that \(\hat I\) and \(\hat\sigma^z\) commute with \(\hat H\) (and listing any other Pauli operators with this property);

(ii) using the result of (a) to argue that any polynomial in \(\hat I, \hat\sigma^z\) is conserved;

(iii) recognising that \(\{\hat I, \hat\sigma^z\}\) already span the diagonal \(2\times 2\) Hermitian matrices, so any conserved observable is of the form \(a_0\hat I + a_z\hat\sigma^z\) — the result of 1.2.2 Problem 1.

(d) State, in one sentence, the physical meaning of the conservation algebra: what does it tell us about the good quantum numbers of a system?