Quantization over quantization

  • Thread starter eljose
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let,s suppose Bohmian mechanics was true then we would have trajectories in the form:

[tex]m\frac{d^{2}x}{dt^2}=-\nabla(V+U_{b}) [/tex] (1) [tex]U_{b}=-\frac{\hbar^2}{2m}\nabla^{2}\psi [/tex] being psi the solution to schroedinguer equation...but the trajectories in (1) comes from the Hamiltonian..
[tex]H=H_0+U_b [/tex] with this we could form the new Schroedinguer equation with function [tex]\psi_{1}(x)[/tex],with new trajectories.....proceeding this infinite times we would have that the real trajectories..(after quantizying infinite times) are:

[tex]m\frac{d^{2}x}{dt^2}=-\nabla(V+U_{total}) [/tex]

with U total [tex]U_{total}=-\frac{\hbar^2}{2m}\nabla(\sum_{n=0}^{\infty}\psi_{n}(x)) [/tex]
 

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  • #2
vanesch
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eljose said:
let,s suppose Bohmian mechanics was true then we would have trajectories in the form:

[tex]m\frac{d^{2}x}{dt^2}=-\nabla(V+U_{b}) [/tex] (1) [tex]U_{b}=-\frac{\hbar^2}{2m}\nabla^{2}\psi [/tex] being psi the solution to schroedinguer equation...but the trajectories in (1) comes from the Hamiltonian..
[tex]H=H_0+U_b [/tex] with this we could form the new Schroedinguer equation with function [tex]\psi_{1}(x)[/tex],with new trajectories.....proceeding this infinite times we would have that the real trajectories..(after quantizying infinite times) are:

[tex]m\frac{d^{2}x}{dt^2}=-\nabla(V+U_{total}) [/tex]

with U total [tex]U_{total}=-\frac{\hbar^2}{2m}\nabla(\sum_{n=0}^{\infty}\psi_{n}(x)) [/tex]

Your approach makes me think of an error a PhD student made (when I was still a lowly undergrad) in applied optics - he was one of my TA, and I couldn't make him see the error. The point is the following:
The mathematical method to obtain a solution of a set of stationary equations (like by doing iterations) HAS NOTHING TO DO WITH ANY SUPPOSED EMERGING DYNAMICS IN TIME :-)
What he did was in fact solving a set of coupled equations:
1) the light propagation through some optical setup
2) that light density affected of course 2 things: 1) population densities of electrons and holes and so on and 2) it heated his apparatus which also changed the optics

He solved his system by iteration: take initial values for populations and temperature, calculate the optical properties, calculate the optics and the light densities, and recalculate from that the adjustments for populations and temperature. And he found a lot of funny (even chaotic) "dynamics" that way, which he mapped through a fudge factor on a real time axis.
Only, for 2) he only used STEADY STATE equations without explicit time derivatives, so my question to him was: how can you hope to get out any DYNAMICAL PICTURE from your static equations ?
And he just replied that it came out of his computer :bugeye:

BTW, to be more to the point: the "quantum potential" in Bohmian mechanics only appears in the equation of motion for the particles and NOT in the potential in the Schroedinger equation, so the iteration doesn't even take place. If you don't like Bohmian mechanics, you can say that it is the fudge term added in Newton's equation to make it come out the same as quantum theory after you already got out the right answer from quantum theory. If you like Bohmian mechanics, you can say that with just a small change in Newton's equation, you succeeded in obtaining all predictions of quantum theory (well, almost :smile:).

cheers,
Patrick.
 

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