NationMaster - Encyclopedia: Ehrenfest theorem

The Ehrenfest theorem, named after Paul Ehrenfest, relates the time derivative of the expectation value for a quantum mechanical operator to the commutator of that operator with the Hamiltonian of the system. It is Paul Ehrenfest Paul Ehrenfest (January 18, 1880 â€“ September 25, 1933) was an Austrian physicist and mathematician from Vienna. ... In mathematics, the derivative is one of the two central concepts of calculus. ... In probability (and especially gambling), the expected value (or expectation) of a random variable is the sum of the probability of each possible outcome of the experiment multiplied by its payoff (value). Thus, it represents the average amount one expects to win per bet if bets with identical odds are... Fig. ... In mathematical formulations of quantum mechanics, an operator is a linear transformation from a Hilbert space to itself. ... For an electrical switch that periodically reverses the current see commutator (electric) In mathematics, the commutator gives an indication of how poorly a certain binary operation fails to be commutative. ... The Hamiltonian, denoted H, has two distinct but closely related meanings. ...

$frac{d}{dt}langle A rangle = frac{i}{hbar}langle [H,A] rangle,$

where A is some QM operator and <A> is its expectation value. Notice how neatly Ehrenfest's theorem fits into the Heisenberg picture of quantum mechanics. The Heisenberg Picture of quantum mechanics is also known as Matrix mechanics. ...

Ehrenfest's theorem is closely related to Liousville's theorem from Hamiltonian mechanics, which involves the Poisson bracket instead of a commutator. In fact, it is a general rule of thumb that a theorem in quantum mechanics which contains a commutator can be turned into a theorem in Classical mechanics by changing the commutator into a Poisson bracket and multiplying by $ihbar$ . In mathematical physics, Liouvilles theorem, named after the French mathematician Joseph Liouville, is a key theorem in classical statistical and Hamiltonian mechanics. ... Hamiltonian mechanics is a re-formulation of classical mechanics that was invented in 1833 by William Rowan Hamilton. ... In mathematics and classical mechanics, the Poisson bracket is an important operator in Hamiltonian mechanics, playing a central role in the definition of the time-evolution of a dynamical system in the Hamiltonian formulation. ... A rule of thumb is an easily learned and easily applied procedure for approximately calculating or recalling some value, or for making some determination. ...

Derivation

Suppose some system is presently in a quantum state $Φ$ . If we want to know the instantaneous time derivative of the expectation value of A, that is, by definition A quantum state is any possible state in which a quantum mechanical system can be. ...

$frac{d}{dt}langle Arangle = frac{d}{dt}int Phi^* A Phi~dV = int left( frac{dPhi^*}{dt} right) APhi~dV + int Phi^* left( frac{dA}{dt}right) Phi~dV +int Phi^* A left( frac{dPhi}{dt} right) ~dV$

$= int left( frac{dPhi^*}{dt} right) APhi~dV + int Phi^* A left( frac{dPhi}{dt} right) ~dV,$

where we are integrating over all space, and we have assumed the operator A is time independant, so that its derivative is zero. If we apply the Schrödinger equation, we find that In physics, the SchrÃ¶dinger equation, proposed by the Austrian physicist Erwin SchrÃ¶dinger in 1925, describes the time-dependence of quantum mechanical systems. ...

$frac{dPhi}{dt} = frac{1}{ihbar}HPhi$

and

$frac{dPhi^*}{dt} = frac{-1}{ihbar}Phi^*H^* = frac{-1}{ihbar}Phi^*H.$ ^[1]

Notice $H = H *$ because the Hamiltonian is hermitian. Placing this into the above equation we have The Hamiltonian, denoted H, has two distinct but closely related meanings. ... A number of mathematical entities are named Hermitian, after the mathematician Charles Hermite: Hermitian matrix Hermitian operator Hermitian adjoint Hermitian form Hermitian metric See also: self-adjoint This is a disambiguation page — a navigational aid which lists other pages that might otherwise share the same title. ...

$frac{d}{dt}langle Arangle = frac{1}{ihbar}int Phi^* (AH-HA) Phi~dV = frac{1}{ihbar}langle [A,H]rangle = frac{i}{hbar}langle[H,A]rangle.$

General example

For the very general example of a massive particle moving in a potential, the Hamiltonian is simply A particle is Look up Particle in Wiktionary, the free dictionary In particle physics, a basic unit of matter or energy. ... It has been suggested that this article or section be merged with Scalar potential. ...

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

where r is just the location of the particle. Suppose we wanted to know the instantaneous change in momentum p. Using Ehrenfest's theorem, we have

$frac{d}{dt}langle prangle = frac{1}{ihbar}langle [p,H]rangle = frac{1}{ihbar}langle [p,V(bold{r})]rangle$

since p commutes with itself. When represented in coordinate space, the momentum operator $p = -ihbarnabla$ , so

$frac{d}{dt}langle prangle = int Phi^* V(bold{r})nablaPhi~dV - int Phi^* nabla (V(bold{r})Phi)~dV.$

After applying a product rule, we have In mathematics, the product rule of calculus, which is also called Leibnizs law (see derivation), governs the differentiation of products of differentiable functions. ...

$frac{d}{dt}langle prangle = langle -nabla V(bold{r})rangle = langle F rangle,$

but we recognize this as Newton's second law. This is an example of the correspondence principle, the result manifests as Newton's second law in the case of having so many particles that the net motion is given exactly by the expectation value of a single particle. Newtons laws of motion are the three scientific laws which Isaac Newton discovered concerning the behaviour of moving bodies. ... In physics, the correspondence principle is a principle, first invoked by Niels Bohr in 1923, which states that the behavior of quantum mechanical systems reduce to classical physics in the limit of large quantum numbers. ...

Notes

^ In Bra-ket notation

$frac{d}{dt}langle phi |xrangle =frac{-1}{ihbar}langle phi |hat{H}|xrangle =frac{-1}{ihbar}langle phi |x rangle H=frac{-1}{ihbar}Phi^*H$

where $hat{H}$ is the Hamiltonian operator, and H is the Hamiltonian represented in coordinate space (as is the case in the derivation above). In other words, we applied the adjoint operation to the entire Schrödinger equation, which flipped the order of operations for H and $Φ$ . Bra-ket notation is the standard notation for describing quantum states in the theory of quantum mechanics. ...

Category: Quantum mechanics

Results from FactBites:

Paul Ehrenfest at AllExperts (2011 words)
	Paul Ehrenfest was born and grew up in Vienna in a Jewish family that was originally from a village in Moravia.
	Ehrenfest's most important contribution in the period from 1912 up to 1933 is the theory of adiabatic invariants.
	Ehrenfest was particularly close to Einstein and to Bohr.

Ehrenfest theorem - Wikipedia, the free encyclopedia (374 words)
	The Ehrenfest theorem, named after Paul Ehrenfest, relates the time derivative of the expectation value for a quantum mechanical operator to the commutator of that operator with the Hamiltonian of the system.
	Ehrenfest's theorem is closely related to Liouville's theorem from Hamiltonian mechanics, which involves the Poisson bracket instead of a commutator.
	The theorem can be shown to follow from the Lindblad equation, a master equation for the time evolution of a mixed state.

More results at FactBites »

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General example

Notes

Derivation