"Brute force" quantization..


by lokofer
Tags: brute force, quantization
lokofer
lokofer is offline
#1
Aug25-06, 10:23 AM
P: 108
Let's suppose we have the "Equation of motions" for a particle:

[tex] F(y'',y',y,x)=0 [/tex] my question is if exsit a "direct" method to apply quantization rules..for example simply stating that:

[tex] F(y'',y',y,x)| \psi (x) >=0 [/tex] or something similar.

- I'm not talking about the usual method (you use the Hamiltonian operator to get the Wave function) but a method to "Quantize" everything without using Hamiltonians or Lagrangians only with the equation of motion and similar...thanks.
Phys.Org News Partner Physics news on Phys.org
Information storage for the next generation of plastic computers
Scientists capture ultrafast snapshots of light-driven superconductivity
Progress in the fight against quantum dissipation
Epicurus
Epicurus is offline
#2
Aug30-06, 09:10 PM
P: 76
The Hellman-feynman theorem deals with the quantum mechanics of forces, but no, there is no procedure which involves quantised forces.
Careful
Careful is offline
#3
Sep5-06, 12:12 PM
P: 1,667
Quote Quote by Epicurus
The Hellman-feynman theorem deals with the quantum mechanics of forces, but no, there is no procedure which involves quantised forces.
Well, I don't know about that theorem, but I was once thinking about quantum mechanics as a probability fluidum (the Madelung interpretation), something I never published since I cannot believe it has not been done yet. In this case, let [tex] R^2 [/tex] be the ``mass´´ density and [tex] \partial_{\mu} S [/tex] be the integrable fluid velocity field. Then, the traditional navier stokes equation is:
[tex] R^2 \partial_t \partial_{\alpha}S + R^2 \partial_{\beta} S \partial_{\beta} \left( \partial_{\alpha} S \right) = \frac{R^2}{m} F_{\alpha} - \partial_{\alpha} p + \partial^{\beta} T_{\beta \alpha} [/tex] and the usual continuity equation
[tex] \partial_t R^2 + \partial^{\alpha} \left( R^2 \partial_{\alpha} S \right) = 0 [/tex]
Now, let the pressure [tex] p = - \frac{1}{2m^2} \left( R \partial_{\beta} \partial^{\beta }R - \frac{1}{3} \partial_{\beta} R \partial^{\beta} R \right) [/tex] and the stress tensor
[tex] T_{\alpha \beta} = - \frac{1}{m^2} \left( \partial_{\alpha}R \partial_{\beta} R - \frac{1}{3} \delta_{\alpha \beta} \partial_{\gamma} R \partial^{\gamma} R \right) [/tex] then it is easy to prove that
with [tex] F_{\alpha} = - \partial_{\alpha} V [/tex], the Navier Stokes equation gives rise to the Hamilton Jacobi equation of Bohmian mechanics. Hence, this provides a general scheme for quantization of particles in general force fields. If you definetly know this has not been done yet, give me a sign and I will post the ``paper'' on the arxiv.

It seems to me you cannot quantize general force fields (in the case of instantaneous action at a distance, there are no travelling waves, hence no particles), only those which can be derived from a (eventually distributional) field theory seem to be meaningful.

Careful


Register to reply

Related Discussions
Difference between "Identical", "Equal", "Equivalent" Calculus & Beyond Homework 9
Difference between "Traction" and "Tractive Force"? General Physics 2
Electrostatic conundrum: How "brute force" and symmetry arg. give different answers! Classical Physics 4