The Heisenberg Picture: Equations of Motion for Operators

Quantum Mechanics A

Schrödinger Equation The most fundamental equation of quantum mechanics; given a Hamiltonian ${\mathcal {H}}$, it describes how a state $|\Psi \rangle$ evolves in time.

Physical Basis of Quantum Mechanics

Basic Concepts and Theory of Motion UV Catastrophe (Black-Body Radiation) Photoelectric Effect Stability of Matter Double Slit Experiment Stern-Gerlach Experiment The Principle of Complementarity The Correspondence Principle The Philosophy of Quantum Theory

Schrödinger Equation

Brief Derivation of Schrödinger Equation Relation Between the Wave Function and Probability Density Stationary States Heisenberg Uncertainty Principle Some Consequences of the Uncertainty Principle

Operators, Eigenfunctions, and Symmetry

Linear Vector Spaces and Operators Commutation Relations and Simultaneous Eigenvalues The Schrödinger Equation in Dirac Notation Transformations of Operators and Symmetry Time Evolution of Expectation Values and Ehrenfest's Theorem

Motion in One Dimension

One-Dimensional Bound States Oscillation Theorem The Dirac Delta Function Potential Scattering States, Transmission and Reflection Motion in a Periodic Potential Summary of One-Dimensional Systems

Discrete Eigenvalues and Bound States

Harmonic Oscillator Spectrum and Eigenstates Analytical Method for Solving the Simple Harmonic Oscillator Coherent States Charged Particles in an Electromagnetic Field WKB Approximation

Time Evolution and the Pictures of Quantum Mechanics

The Heisenberg Picture: Equations of Motion for Operators The Interaction Picture The Virial Theorem

Angular Momentum

Commutation Relations Angular Momentum as a Generator of Rotations in 3D Spherical Coordinates Eigenvalue Quantization Orbital Angular Momentum Eigenfunctions

Central Forces

General Formalism Free Particle in Spherical Coordinates Spherical Well Isotropic Harmonic Oscillator Hydrogen Atom WKB in Spherical Coordinates

The Path Integral Formulation of Quantum Mechanics

Feynman Path Integrals The Free-Particle Propagator Propagator for the Harmonic Oscillator

Continuous Eigenvalues and Collision Theory

Differential Cross Section and the Green's Function Formulation of Scattering Central Potential Scattering and Phase Shifts Coulomb Potential Scattering

There are several ways to mathematically approach the change of a quantum mechanical system with time. The Schrödinger picture considers wave functions which change with time while the Heisenberg picture places the time dependence on the operators and deals with wave functions that do not change in time. The interaction picture, also known as the Dirac picture, is somewhere between the other two, placing time dependence on both operators and wave functions. The two "pictures" correspond, theoretically, to the time evolution of two different things. Mathematically, all methods should produce the same result.

Definition of the Heisenberg Picture

The time evolution operator is at the heart of the "evolution" of a state, and is the same in each picture. The evolution operator is obtained from the time dependent Schrödinger equation after separating the time and spatial parts of the wave equation:

$i\hbar {\frac {\partial }{\partial t}}U(t,t_{0})=HU(t,t_{0})$

The solution to this differential equation depends on the form of .

If we know the time evolution operator, $U$, and the initial state of a particular system, all that is needed is to apply to the initial state ket. We then obtain the ket for some later time.

$U|\alpha (0)\rangle =|\alpha (t)\rangle .$

Therefore, if we know the initial state of a system, we can obtain the expectation value of an operator, A, at some later time:

$\langle A\rangle (t)=\langle \alpha (t)|A|\alpha (t)\rangle =\langle \alpha (0)|U^{\dagger }AU|\alpha (0)\rangle .$

We can make a redefinition by claiming that

$A_{H}(t)=U^{\dagger }AU$

and taking as our state kets the time independent, initial valued state ket $|\alpha (0)\rangle$.

This formulation of quantum mechanics is called the Heisenberg picture. In this picture, operators evolve in time and state kets do not. (Note that the difference between the two pictures only lies in the way we write them down).

In classical physics we obviously have time evolution of a system - our observable (position or angular momentum or whatever variable we are considering) changes in a way dictated by the classical equations of motion. We do not talk about state kets in classical mechanics. Therefore, the Heisenberg, where the operator changes in time, is useful because we can see a closer connection to classical physics than with the Schrödinger picture.

Comparing the Heisenberg Picture and the Schrödinger Picture

As mentioned above, both the Heisenberg representation and the Schrödinger representation give the same results for the time dependent expectation values of operators.

In the Schrödinger picture, in which we are most accustomed, the states change over time while the operators remain constant. In other words, the time dependence is carried by the state operators. When the Hamiltonian is independent of time, it is possible to write:

$|\Psi (0)\rangle$

as the state at t = 0.

At another time t, this becomes

$|\Psi (t)\rangle =e^{-iHt/\hbar }|\Psi (0)\rangle$,

which in this form solves the Schrödinger equation

$i\hbar {\frac {\partial |\Psi (t)\rangle }{\partial t}}=H|\Psi (t)\rangle$.

Conversely, in the Heisenberg picture to find the time dependent expectation of a given operator, the states remain constant while the operators change in time. In this case, you can write the time dependent operator as:

$A(t)=e^{iHt/\hbar }Ae^{-iHt\hbar }$,

which means the expectation value is,

$\langle A\rangle _{t}=\langle \Psi (0)|A(t)|\Psi (0)\rangle$.

The Heisenberg Equation of Motion

In the Heisenberg picture, the quantum mechanical observables change in time as dictated by the Heisenberg equations of motion. We can study the evolution of a Heisenberg operator by differentiating equation 2 with respect to time:

${\begin{aligned}{\frac {d}{dt}}A_{H}&={\frac {\partial U^{\dagger }}{\partial t}}AU+U^{\dagger }{\frac {\partial A}{\partial t}}U+U^{\dagger }A{\frac {\partial U}{\partial t}}\\&={\frac {-1}{i\hbar }}U^{\dagger }HUU^{\dagger }AU+\left({\frac {\partial A}{\partial t}}\right)_{H}+U^{\dagger }AUU^{\dagger }HU{\frac {1}{i\hbar }}\\&={\frac {1}{i\hbar }}\left(\left[A,H\right]\right)_{H}+\left({\frac {\partial A}{\partial t}}\right)_{H}.\end{aligned}}$

The last equation is known as the Ehrenfest's Theorem.

For example, if we have a hamiltonian of the form,

$H={\frac {p^{2}}{2m}}+V(r),$

then we can find the Heisenberg equations of motion for p and r.

The position operator in 3D is: $i\hbar {\frac {d{\hat {r}}(t)}{dt}}=\left[{\hat {r}}(t),H\right]$

Since the Hamiltonian as given above has a V(r,t) term and a momentum term the two commuators are as follows:

$\left[{\hat {r}}(t),V(r,t)\right]=0$

$\left[{\hat {r}}(t),{\frac {p(t)^{2}}{2m}}\right]={\frac {i\hbar p(t)}{m}}$

this yields ${\frac {d{\hat {r}}(t)_{H}}{dt}}={\frac {{\hat {p}}(t)_{H}}{m}}$.

To find the equations of motion for the momentum you need to evaluate $\left[{\hat {p}}(t),V(r(t))\right]$,

which equals, $\left[{\hat {p}},V(r)\right]=-i\hbar \nabla V(r)$.

This yields ${\frac {d{\hat {p}}_{H}}{dt}}=-\nabla V({\hat {r}}_{H}).$

These results are, of course, what we would expect if we only knew classical physics, providing another example of how the Heisenberg picture is more transparently similar to classical physics.

In particular, if we apply these equations to the Harmonic oscillator with natural frequency $\omega ={\sqrt {\frac {k}{m}}}\!$

${\frac {d{\hat {x}}_{H}}{dt}}={\frac {{\hat {p}}_{H}}{m}}$

${\frac {d{\hat {p}}_{H}}{dt}}=-k{{\hat {x}}_{H}}.$

we can solve the above equations of motion and find

${\hat {x}}_{H}(t)={\hat {x}}_{H}(0)\cos(\omega t)+{\frac {{\hat {p}}_{H}(0)}{m\omega }}\sin(\omega t)$

${\hat {p}}_{H}(t)={\hat {p}}_{H}(0)\cos(\omega t)-{\hat {x}}_{H}(0)m\omega \sin(\omega t).\!$

It is important to stress that the above oscillatory solution is for the position and momentum operators. Also, note that ${\hat {x}}_{H}(0)\!$ and ${\hat {p}}_{H}(0)\!$ correspond to the time independent operators ${\hat {x}}\!$ and ${\hat {p}}\!$.