Time Evolution of Expectation Values and Ehrenfest's Theorem

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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.
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
Brief Derivation of Schrödinger Equation
Relation Between the Wave Function and Probability Density
Stationary States
Heisenberg Uncertainty Principle
Some Consequences of the Uncertainty Principle
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
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
Harmonic Oscillator Spectrum and Eigenstates
Analytical Method for Solving the Simple Harmonic Oscillator
Coherent States
Charged Particles in an Electromagnetic Field
WKB Approximation
The Heisenberg Picture: Equations of Motion for Operators
The Interaction Picture
The Virial Theorem
Commutation Relations
Angular Momentum as a Generator of Rotations in 3D
Spherical Coordinates
Eigenvalue Quantization
Orbital Angular Momentum Eigenfunctions
General Formalism
Free Particle in Spherical Coordinates
Spherical Well
Isotropic Harmonic Oscillator
Hydrogen Atom
WKB in Spherical Coordinates
Feynman Path Integrals
The Free-Particle Propagator
Propagator for the Harmonic Oscillator
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.

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