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Brute force quantization.. 
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#1
Aug2506, 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. 


#2
Aug3006, 09:10 PM

P: 76

The Hellmanfeynman theorem deals with the quantum mechanics of forces, but no, there is no procedure which involves quantised forces.



#3
Sep506, 12:12 PM

P: 1,667

[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 


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