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Physics LibreTexts

3.4: Ehrenfest's Theorem

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A simple way to calculate the expectation value of momentum is to evaluate the time derivative of x, and then multiply by the mass m: that is,

p=mdxdt=mddtx|ψ|2dx=mx|ψ|2tdx.

However, it is easily demonstrated that

|ψ|2t+jx=0

[this is just the differential form of Equation ([epc])], where j is the probability current defined in Equation ([eprobc]). Thus,

p=mxjxdx=mjdx,

where we have integrated by parts. It follows from Equation ([eprobc]) that

p=i2(ψψxψxψ)dx=iψψxdx,

where we have again integrated by parts. Hence, the expectation value of the momentum can be written

p=mdxdt=iψψxdx.

It follows from the previous equation that dpdt=i(ψtψx+ψ2ψtx)dx=[(iψt)ψx+ψx(iψt)]dx,

where we have integrated by parts. Substituting from Schrödinger’s equation ([e3.1]), and simplifying, we obtain dpdt=[22mx(ψxψx)+V(x)|ψ|2x]dx=V(x)|ψ|2xdx.
Integration by parts yields dpdt=dVdx|ψ|2dx=dVdx.

Hence, according to Equations ([e4.34x]) and ([e3.41]),

mdxdt=p,dpdt=dVdx.

Evidently, the expectation values of displacement and momentum obey time evolution equations that are analogous to those of classical mechanics. This result is known as Ehrenfest’s theorem .

Suppose that the potential V(x) is slowly varying. In this case, we can expand dV/dx as a Taylor series about x. Keeping terms up to second order, we obtain

dV(x)dx=dV(x)dx+dV2(x)dx2(xx)+12dV3(x)dx3(xx)2.

Substitution of the previous expansion into Equation ([e3.43]) yields dpdt=dV(x)dxσ2x2dV3(x)dx3,

because 1=1, and xx=0, and (xx)2=σ2x. The final term on the right-hand side of the previous equation can be neglected when the spatial extent of the particle wavefunction, σx, is much smaller than the variation length-scale of the potential. In this case, Equations ([e3.42]) and ([e3.43]) reduce to mdxdt=p,dpdt=dV(x)dx.
These equations are exactly equivalent to the equations of classical mechanics, with x playing the role of the particle displacement. Of course, if the spatial extent of the wavefunction is negligible then a measurement of x is almost certain to yield a result that lies very close to x. Hence, we conclude that quantum mechanics corresponds to classical mechanics in the limit that the spatial extent of the wavefunction (which is typically of order the de Boglie wavelength) is negligible. This is an important result, because we know that classical mechanics gives the correct answer in this limit.

Contributors and Attributions

  • Richard Fitzpatrick (Professor of Physics, The University of Texas at Austin)


This page titled 3.4: Ehrenfest's Theorem is shared under a not declared license and was authored, remixed, and/or curated by Richard Fitzpatrick.

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