Ehrenfest=s Theorem

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    One of the tests that a theory must pass before we incorporate it into the science of physics necessitates that under the right circumstances that theory must coincide with the classical physics of Isaac Newton. Niels Henrik David Bohr (1885 Oct 07 - 1962 Nov 18) presented this test as the correspondence principle and applied it to quantum theory. For example, if we let the speed of light increase toward infinity (that nonexistent place at the Aend@ of the endless number line), then the Lorentz factor approaches one as a limit and the equations of relativistic dynamics revert to their Newtonian equivalents. As for the quantum theory, the controlling parameter is Planck=s constant and that must go to zero to create a Newtonian universe; that is,

(Eq=n 1)

With Relativity the transformation to classical physics occurs directly, but with the quantum theory the formalism differs from that of classical physics to such a degree that we have a bit of a quandary as to how we should proceed.

    Austrian physicist Paul Ehrenfest (1880 Jan 18 - 1933 Sep 25) devised a way to make the transformation. He simply calculated the time derivatives of the expectation values of a particle= s position and linear momentum. For the particle=s position we have

(Eq=n 2)

in which we integrate over the entire span of the x-direction, the other two dimensions being divided out of the element of volume because they have no relevance to this particular calculation. To carry that calculation forward we take Schrödinger=s Equation,

(Eq=n 3)

and use it to substitute for the time derivatives in Equation 2. In this case we assume that the potential energy U is not a function of position in the x-direction. Equation 2 thus becomes

(Eq=n 4)

Note that the terms involving the potential energy canceled each other out. We can integrate the first term on the right side of the equality sign by parts:

(Eq=n 5)

The first term on the right side of that equation, when evaluated at the limits of the integral=s range, vanishes because the product xΨ must go to zero at the boundary of space to ensure that the particle has no possibility of existence there. Integrating the surviving term by parts gives us

(Eq=n 6)

In this case the first term on the right side of the equality sign vanishes because psi-conjugate must go to zero on the boundary of space for the same reason as given above. So now we have Equation 4 as

(Eq=n 7)

In the last step of that calculation I used the fact that the operator

(Eq=n 8)

extracts all relevant values of the x-component of the particle= s linear momentum from the state function and thereby calculated the expectation value of that property. We would have obtained the same result if I had integrated the second term on the right side of Equation 4 by parts. So we see that the time derivative of the particle=s expected position on the x-axis equals the expectation value of the particle=s x-ward linear momentum divided by the particle=s mass, which corresponds to the classical case.

    Next we want to calculate the time derivative of the expectation value of the particle=s x-ward linear momentum,

(Eq=n 9)

So we have

(Eq=n 10)

As we did in Equations 5 and 6, we integrate the first term in the first integral on the right twice by parts and find that the whole integral cancels out. Noting that in this case the potential must be a function of the particle=s position in the x-direction (U=U(x)), we thus have

(Eq=n 11)

So the expectation value of the negative gradient of the particle=s potential energy corresponds to the rate at which the expectation value of the particle=s linear momentum changes in time. Again, that coincides with the classical case.

    But all of the above is based on Schrödinger=s classical wave equation. In my version of the quantum theory we begin using the relativistic version immediately, so we must now ask whether Ehrenfest=s theorem applies to the relativistic quantum theory.

    For a particle whose only properties consist of mass and a force-engaging charge we use Schrödinger=s relativistic wave equation, also known as the Klein-Gordon Equation,

(Eq=n 12)

which we obtain by applying the operators

(Eq=n 13)


(Eq=n 14)

to the relativistic relationship between kinetic energy and linear momentum

(Eq=n 15)

    To include the potential energy of the particle in that equation we cannot simply add it in. In the relativistic description of a particle its dynamic properties must appear as four-vectors, four-dimensional vectors that differ between inertial frames in accordance with the Lorentz Transformation. In the case of linear momentum, the kinetic energy plays the role of the timeward component. On that basis we identify the particle= s potential energy (qφ) as the temporal component of the four potential and the corresponding vector potential (qA, the potential momentum) as the spatial components. Equation 15 then becomes

(Eq=n 16)

In this case the wave function has the form

(Eq=n 17)

    Now we want to see whether that equation gives us a description of Reality that conforms to Niels Bohr=s statement that, AEvery description of natural processes must be based on ideas which have been introduced and defined by the classical theory.@ When we extend our study of physics into realms beyond human experience, to speeds close to the speed of light or to actions on the atomic scale, we must necessarily adopt and use concepts that we have developed for the physics of phenomena that we study at the human scale. Because we expect Reality to consist of a continuum, we expect that the laws of physics that we devise to describe the extra-human realms will segue into the familiar laws of physics as we bring them into the human-scale realm, as by making some parameter approach a limiting value. Now we want to see whether the relativistic quantum theory bears that ætherial watermark. To that end we want to calculate the expectation value of the rate at which a particle=s linear momentum changes with the elapse of time.

    Using the relativistic formula for the probability density, we have

(Eq=n 18)

In that equation the operators extract the argument of the wave function and differentiate it, so we have

(Eq=n 19)

The vector variables x and p do not represent fields, but rather represent points in phase space that the particle occupies as time elapses, so we take the spatial derivatives of those variables as equal to zero. Further, we didn=t want to introduce the complications of radiation fields, so with respect to the source of the potential fields we must have dφ/dt=0 and dA/dt=0. Carrying out the differentiations thus gives us

(Eq=n 20)

Substituting that result and its complex conjugate into Equation 18 then gives us

(Eq=n 21)

which describes the electromagnetic force plus the force due to any other static potentials the particle encounters. Thus we gain strong evidence that the relativistic quantum theory, like its non-relativistic counterpart, satisfies Bohr=s correspondence principle.


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