Time Evolution of Expectation Values and Ehrenfest's Theorem

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

Time Evolution of Expecation Values

Having described in a previous section how the state vector of a system evolves in time, we may now derive a formula for the time evolution of the expectation value of an operator. Given an operator ${\hat {O}}(t),$ we know that its expectation value is given by $\langle {\hat {O}}(t)\rangle =\langle \Psi (t)|{\hat {O}}(t)|\Psi (t)\rangle .$ If we take the time derivative of this expectation value, we get

${\frac {d\langle {\hat {O}}(t)\rangle }{dt}}=\left[{\frac {d}{dt}}\langle \Psi (t)|\right]{\hat {O}}(t)|\Psi (t)\rangle +\langle \Psi (t)|{\frac {d{\hat {O}}(t)}{dt}}|\Psi (t)\rangle +\langle \Psi (t)|{\hat {O}}(t)\left[{\frac {d}{dt}}|\Psi (t)\rangle \right]$

$=\left[{\frac {d}{dt}}\langle \Psi (t)|\right]{\hat {O}}(t)|\Psi (t)\rangle +\langle \Psi (t)|{\hat {O}}(t)\left[{\frac {d}{dt}}|\Psi (t)\rangle \right]+\left\langle {\frac {d{\hat {O}}(t)}{dt}}\right\rangle .$

We now use the Schrödinger equation and its dual to write this as

${\frac {d\langle {\hat {O}}(t)\rangle }{dt}}={\frac {i}{\hbar }}\langle \Psi (t)|{\hat {H}}(t){\hat {O}}(t)|\Psi (t)\rangle -{\frac {i}{\hbar }}\langle \Psi (t)|{\hat {O}}(t){\hat {H}}(t)|\Psi (t)\rangle +\left\langle {\frac {d{\hat {O}}(t)}{dt}}\right\rangle$

$={\frac {i}{\hbar }}\langle [{\hat {H}}(t),{\hat {O}}(t)]\rangle +\left\langle {\frac {d{\hat {O}}(t)}{dt}}\right\rangle .$

This formula is of the utmost importance in all facets of quantum mechanics.

Ehrenfest's Theorem

We now use the above result to prove Ehrenfest's Theorem, which states that the expecation values of the position and momentum operators obey the same equations that the corresponding classical quantities obey. Thus, one may consider this theorem to be a manifestation of the correspondence principle, because the expectation values, in the classical limit, become the values of the corresponding classical quantities.

Consider the Hamiltonian,

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

We are now interested in determining how the expecation values of the position ${\hat {\mathbf {r} }}$ and momentum ${\hat {\mathbf {p} }}$ operators evolve in time. Using the formula that we just derived, and noting that neither operator depends explicitly on time, we obtain

${\frac {d\langle {\hat {\mathbf {r} }}\rangle }{dt}}={\frac {i}{\hbar }}\langle [{\hat {H}},{\hat {\mathbf {r} }}]\rangle$ and ${\frac {d\langle {\hat {\mathbf {p} }}\rangle }{dt}}={\frac {i}{\hbar }}\langle [{\hat {H}},{\hat {\mathbf {p} }}]\rangle .$

Using the fact that $[{\hat {p}}_{i}^{2},{\hat {x}}_{j}]=-2i\hbar p_{i}\delta _{ij}$ and $[p_{i},f({\hat {\mathbf {r} }})]=-i\hbar {\frac {\partial f({\hat {\mathbf {r} }})}{\partial {\hat {x}}_{i}}},$ we find that

${\frac {d\langle {\hat {\mathbf {r} }}\rangle }{dt}}={\frac {\langle {\hat {\mathbf {p} }}\rangle }{m}}$ and ${\frac {d\langle {\hat {\mathbf {p} }}\rangle }{dt}}=-\langle \nabla V({\hat {\mathbf {r} }})\rangle .$

These two equations closely resemble equations familar from classical mechanics - the first resembles the statement that momentum is equal to mass times velocity, while the latter looks like Newton's second law.