Stephen Paul King (stephenk1@home.com)
Thu, 10 Jun 1999 16:26:17 GMT
On 9 Jun 1999 15:11:14 GMT, hunk@alpha1.csd.uwm.edu (Mark William
Hopkins) wrote:
>THEOREM:
> A particle satisfying the non-relativistic Schroedinger equation is
>equivalently characterized as a fluid with the following properties
>
> (1) It is irrotational
> (2) It occupies all space
> (3) It has finite total mass
> (4) It satisfies the Continuity Equation
> (5) It satisfies the Euler Equation
> (6) It has a non-isotropic pressure satisfying the Equation of State:
>
> / h-bar \ 2 / (del rho) (del rho) \
> P = | ----- | | ------------------- - del^2 rho I |
> \ 2m / \ rho /
>where
> P = the pressure dyad
> m = the total mass of the fluid
> rho = the mass density of the fluid
> del = the vector gradient operator
>
>Dyadic notation was used above with
> I = the identity dyad
> u v = the linear operator which maps vector w |-> u (v.w)
>
>This pressure has the following principle components:
>
> p1 = -(h-bar/2m)^2 rho del^2 (ln(rho)) parallel to (del rho)
> p2 = -(h-bar/2m)^2 del^2 rho perpendicular to (del rho)
>
>The pressure is isotropic at the stationary points of rho and is
>approximately isotropic wherever the density is slowly varying. For
>sinusoidally varying densities, the longitudinal pressure will be
>constant, and the pressure in transverse directions will satisfy the
>Bode law for perfect gases with a speed of sound given by
>
> c_sound = h-bar k / 2m k = wavenumber
>
>This provides us with an interesting interpretation for the wavelength
>
> lambda_c = 2mc/h
>
>Proof:
> The key is the Euler equation, stated in generalized form for pressure
>tensors as follows:
> @v/@t + v.del v = F/m - (div P)/rho
>
>where @ is being used as the ASCII rendition of the partial derivative
>operator, where v represents the velocity field of the fluid. For
>isotropic pressures (i.e., P = p I), the dyad expression (div P)
>reduces to the more familiar form (del p).
>
> The fluid, by assumption is irrotational, which means that
>
> curl v = 0.
>
>Under these conditions, it is generally true that
>
> v.del v = del (v^2/2)
>
> The force F, by assumption, is conservative, so that it has a potential
>V for which F = -del V. That allows us to write
>
> m @v/@t + del (m v^2/2 + V) = -(div P)/A^2
>where
> A = the 'amplitude' of the fluid, defined by
> A^2 = rho/m
>
>The quantity A is what is interpreted, under the Copenhagen Interpretation,
>as the probability amplitude of the particle's wave function, apart from
>a complex phase factor.
>
> If you work out -(div P)/(A^2), you will get
>
> div P
> - ----- = del Q
> A^2
>where
> h-bar^2 del^2 A
> Q = - ------- -------
> 2m A
>
>is none other than Bohm's infamous Quantum Potential.
>
> This simple, but rather unexpected, conversion of P to Q is the secret.
>For, just as in the discussions carried out by Bohm in his famous
>hidden-variables paper, we note that the one and only factor which marks the
>departure from Classical Physics is the appearance in the Euler Equation of
>the extra term corresponding to the universal force:
>
> h-bar^2 / del^2 A \
> F = -del Q = ------- del | ------- |
> 2m \ A /
>
>generated by the energy field whose expression is given by Q, Bohm's
>Quantum Potential.
>
> Thus, the Euler equation reduces to the form
>
> m @v/@t + del E = 0
>where
> E = m v^2/2 + V + Q
>
>may be interpreted as the local energy of the fluid.
>
>Since the velocity field is irrotational, this equation allows us to
>derive it from a potential, S, satisfying the equations
>
> del S = p = m v
> > -@S/@t = E = m v^2/2 + V + Q
>
>where p may be interpreted as the local momentum of the fluid. We also note
>that since A^2 is proportional to rho, we may rewrite the Continuity Equation
>in the form:
>
> @(A^2)/@t + div (A^2 v) = 0
>
>From here, we join company with Bohm's developments, which proceed upon
>the identification of the wavefunction psi as
>
> psi = A exp(i S/h-bar)
>
>With these identifications, the defining characteristics listed in the
>theorem may be recovered, thus establishing the equivalence.
>
>-----------
>
>A historical note. It may be of interest to note how this development
>arose, since its origin is entirely independent of Bohm, and since the
>continuation of the original line of development may be something worth
>pursuing in itself.
>
>As any student of Quantum Physics knows, the Schroedinger equation yields
>a continuity equation for certain quantities, derived from the wavefunction,
>which have natural interpretation as a kind of flow. The nature of this
>flow was actually the focus of the debate in the 1920's that led ultimately
>to the Copenhagen Interpretation.
>
>An interesting note about the interpretation implied by the theorem above,
>aside from the fact that it is the more natural, intuitive interpretation
>(and probably the one that the Copenhagen people had originally settled
>upon at first) is that is actually more general. The Euler equation
>could also be stated for forces that are non-conservative. There is
>no direct generalization of the Schroedinger equation that I'm aware of
>that applies to non-conservative forces. But this has experimental
>consequences, namely:
>
> Is the generalization of the abovementioned
>
> Euler equation + Continuity Equation + Equation of State
>
>to non-conservative forces experimentally valid? As an exercise, try
>converting this more general system into a form involving the wavefunction
>psi, to see what it looks like in more customary notation.
>
>Anyway, what prompted these developments were the following questions:
>
> Can the Schroedinger Equation by solved by analog
> computation using an actual fluid?
>
> If so, what properties would that fluid have to satisfy?
>
>The answer is that stated in the theorem, itself:
>
> The fluid must be irrotational and must satisfy a very unusual
> (but rather simple, elegant, and vaguely familiar) equation of state.
>
>Undoubtedly, the irrotational nature of the fluid has a direct connection
>to the fact that particle spin is absent from the Schroedinger equation.
>
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