The Physics Travel Guide

Heisenberg Equation

Prerequisites: Canonical Quantum Mechanics. Full path (17 pages)

dF^dt=−iℏ[F^,H^]+∂F^∂t\frac{\mathrm{d}\hat F}{\mathrm{d}t} = -\frac{i}{\hbar}[\hat F,\hat H] + \frac{\partial \hat F}{\partial t}

Intuitive

There are two equivalent ways to describe how a quantum system changes in time. In the first, the state of the system changes, while the quantities we can measure (position, momentum, energy, spin) are fixed. This is the picture behind the Schrödinger equation. In the second, the state stays fixed once and for all, and the measurable quantities themselves change with time. The Heisenberg equation is the equation of motion in this second picture. Both pictures give the same predictions for every measurement, in the same way that we can describe a rotation either by turning an object or by turning the coordinate system in the opposite direction.

The Heisenberg picture is closer to classical mechanics. There, we also ask how position and momentum change in time, and the Heisenberg equation for the position and momentum operators has the same form as Newton's equations. What makes it quantum is that the operators do not commute. The equation says that a quantity changes in time to the extent that it fails to commute with the energy of the system. A quantity that commutes with the energy, and does not depend on time explicitly, stays constant. This gives a direct way to find conserved quantities.

Concrete

Derivation from the Schrödinger picture. For a Hamiltonian H^\hat H that does not depend on time, the state evolves as ∣ψ(t)⟩=U^(t)∣ψ(0)⟩|\psi(t)\rangle = \hat U(t)|\psi(0)\rangle with U^(t)=e−iH^t/ℏ\hat U(t) = e^{-i\hat H t/\hbar}. The expectation value of an observable F^\hat F is

⟨ψ(t)∣F^∣ψ(t)⟩=⟨ψ(0)∣ U^†(t)F^ U^(t) ∣ψ(0)⟩.\langle \psi(t)|\hat F|\psi(t)\rangle = \langle \psi(0)|\,\hat U^\dagger(t)\hat F\,\hat U(t)\,|\psi(0)\rangle .

We can therefore put the time dependence into the operator, F^H(t)=U^†(t)F^U^(t)\hat F_H(t) = \hat U^\dagger(t)\hat F\hat U(t), and keep the state fixed. Using ddtU^=−iℏH^U^\frac{d}{dt}\hat U = -\frac{i}{\hbar}\hat H\hat U and ddtU^†=iℏU^†H^\frac{d}{dt}\hat U^\dagger = \frac{i}{\hbar}\hat U^\dagger\hat H, and the fact that H^\hat H commutes with U^\hat U,

dF^Hdt=iℏ(H^F^H−F^HH^)+U^†∂F^∂tU^=−iℏ[F^H,H^]+(∂F^∂t)H,\frac{d\hat F_H}{dt} = \frac{i}{\hbar}\left(\hat H\hat F_H - \hat F_H\hat H\right) + \hat U^\dagger\frac{\partial\hat F}{\partial t}\hat U = -\frac{i}{\hbar}[\hat F_H,\hat H] + \left(\frac{\partial \hat F}{\partial t}\right)_H ,

which is the equation in the box above. The last term only appears if the operator depends on time explicitly, for example through an external field.

Example. For a particle with H^=p^22m+V(x^)\hat H = \frac{\hat p^2}{2m} + V(\hat x) and [x^,p^]=iℏ[\hat x,\hat p] = i\hbar we need [x^,p^2]=2iℏp^[\hat x,\hat p^2] = 2i\hbar\hat p and [V(x^),p^]=iℏV′(x^)[V(\hat x),\hat p] = i\hbar V'(\hat x). Then

dx^dt=−iℏ[x^,H^]=p^m,dp^dt=−iℏ[p^,H^]=−V′(x^).\frac{d\hat x}{dt} = -\frac{i}{\hbar}[\hat x,\hat H] = \frac{\hat p}{m}, \qquad \frac{d\hat p}{dt} = -\frac{i}{\hbar}[\hat p,\hat H] = -V'(\hat x).

These have the form of Hamilton's equations, now for operators. Taking expectation values gives Ehrenfest's theorem, md2dt2⟨x^⟩=−⟨V′(x^)⟩m\frac{d^2}{dt^2}\langle \hat x\rangle = -\langle V'(\hat x)\rangle. For the harmonic oscillator, V=12mω2x^2V = \frac12 m\omega^2\hat x^2, the equations are linear and we can solve them exactly:

x^(t)=x^(0)cos⁡ωt+p^(0)mωsin⁡ωt.\hat x(t) = \hat x(0)\cos\omega t + \frac{\hat p(0)}{m\omega}\sin\omega t .

Conserved quantities. If [F^,H^]=0[\hat F,\hat H] = 0 and F^\hat F has no explicit time dependence, then F^H(t)=F^\hat F_H(t) = \hat F is constant. In particular H^\hat H itself is conserved.

Relation to classical mechanics. In the Hamiltonian formalism a function F(q,p,t)F(q,p,t) evolves as dFdt={F,H}+∂F∂t\frac{dF}{dt} = \{F,H\} + \frac{\partial F}{\partial t} with the Poisson bracket {⋅,⋅}\{\cdot,\cdot\}. The Heisenberg equation has the same form with {F,H}\{F,H\} replaced by 1iℏ[F^,H^]\frac{1}{i\hbar}[\hat F,\hat H]. This is the correspondence that canonical quantization is built on.

Abstract

Let H\mathcal H be a Hilbert space and H^\hat H a self-adjoint operator on it, possibly unbounded. By Stone's theorem, U^(t)=e−iH^t/ℏ\hat U(t) = e^{-i\hat H t/\hbar} is a strongly continuous one-parameter group of unitary operators, and every such group arises this way. The Heisenberg dynamics is the one-parameter group of automorphisms

αt(F^)=U^(t)† F^ U^(t)\alpha_t(\hat F) = \hat U(t)^\dagger\,\hat F\,\hat U(t)

of the algebra of bounded operators B(H)\mathcal B(\mathcal H). Its generator is the derivation δ(F^)=iℏ[H^,F^]\delta(\hat F) = \frac{i}{\hbar}[\hat H,\hat F], and the Heisenberg equation is ddtαt(F^)=αt(δ(F^))\frac{d}{dt}\alpha_t(\hat F) = \alpha_t(\delta(\hat F)). For unbounded H^\hat H the commutator is only defined on a suitable domain, and the equation holds there in the strong sense, while the automorphism group itself is always well defined.

This formulation makes sense without a Hilbert space. In the algebraic approach, a system is described by a C∗C^*-algebra A\mathfrak A of observables, and dynamics is a one-parameter group of ∗*-automorphisms αt\alpha_t of A\mathfrak A. A state ω\omega gives the expectation values ω(αt(F^))\omega(\alpha_t(\hat F)). The Schrödinger picture corresponds to transporting the state, ωt=ω∘αt\omega_t = \omega\circ\alpha_t, which gives the same numbers. This is useful in quantum statistical mechanics and quantum field theory, where the Hamiltonian of an infinite system need not exist as an operator on a given representation while the automorphism group still does.

For a time-dependent Hamiltonian, U^(t)\hat U(t) is the time-ordered exponential of −iℏ∫0tH^(t′) dt′-\frac{i}{\hbar}\int_0^t \hat H(t')\,dt', the automorphisms no longer form a group, and the commutator in the Heisenberg equation involves H^H(t)=U^†(t)H^(t)U^(t)\hat H_H(t) = \hat U^\dagger(t)\hat H(t)\hat U(t).

The structure parallels classical mechanics on a symplectic manifold, where the Hamiltonian vector field XHX_H generates a flow ϕt\phi_t and observables evolve by pullback, F↦F∘ϕtF\mapsto F\circ\phi_t, with generator {⋅,H}\{\cdot,H\}. The map {⋅,⋅}→1iℏ[⋅,⋅]\{\cdot,\cdot\}\to\frac{1}{i\hbar}[\cdot,\cdot] cannot be realised exactly as a Lie algebra homomorphism on all polynomial observables with qq and pp acting irreducibly (Groenewold-van Hove theorem). Deformation quantization realises it to leading order in ℏ\hbar, with the Moyal bracket in place of the commutator.

Why is it interesting?

The Heisenberg equation describes the time-evolution of observables in quantum mechanics.

Path to this page

  1. Calculus
  2. Imaginary Numbers
  3. Linear Algebra
  4. Functional
  5. Hilbert Space
  6. Legendre Transformation
  7. Newton's Second Law
  8. Vector Calculus
  9. Newtonian Mechanics
  10. Variational Calculus
  11. Configuration Space
  12. Euler-Lagrange Equations
  13. Lagrangian Formalism
  14. Phase Space
  15. Hamiltonian Formalism
  16. Schrödinger Equation
  17. Canonical Quantum Mechanics