**Stephen Paul King** (*stephenk1@home.com*)

*Thu, 10 Jun 1999 16:26:17 GMT*

**Messages sorted by:**[ date ] [ thread ] [ subject ] [ author ]**Next message:**Stephen P. King: "[time 402] Re: [time 400] On the Problem of Information Flow between LSs"**Previous message:**Matti Pitkanen: "[time 400] Re: [time 399] On the Problem of Information Flow between LSs"**In reply to:**Stephen P. King: "[time 399] On the Problem of Information Flow between LSs"

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
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*> 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
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*>sinusoidally varying densities, the longitudinal pressure will be
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*>constant, and the pressure in transverse directions will satisfy the
*

*>Bode law for perfect gases with a speed of sound given by
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*>
*

*> 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.
*

*>
*

**Next message:**Stephen P. King: "[time 402] Re: [time 400] On the Problem of Information Flow between LSs"**Previous message:**Matti Pitkanen: "[time 400] Re: [time 399] On the Problem of Information Flow between LSs"**In reply to:**Stephen P. King: "[time 399] On the Problem of Information Flow between LSs"

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