- #1

eljose

- 492

- 0

let be the solution to SE in the form [tex]\psi=exp(iS/\hbar)[/tex] where S has the "exact" differential equation solution in the form:

[tex]\frac{dS}{dt}+\frac{1}{2m}(\nabla{S})^{2}+V(x)-\frac{i\hbar}{2m}(\nabla^{2}{S}) [/tex]

then we could form the complex potential:[tex]U=V(x)-\frac{i\hbar}{2m}(\nabla^{2}{S})[/tex]

and the Classical equation of Motion in the form:

[tex]m\frac{d^{2}x}{dx^{2}}=-\nabla{U} [/tex]

How do we solve equation (1)?...for example we use a trial function for S=f(r,t) then we calculate [tex]\nabla^{2}{f(r,t)[/tex] and introduce it into equation (1),solve S for this function f(r,t) and again we introduce into the differential equation to find another value of S more accurate than before.

Complex trajectories...are they allowed?..remember that the particle can be into a "classical forbidden" region,then if we use the eikonal equation of Optics [tex](\nabla{S})^{2}=n^{2} [/tex] with n the refraction index of the material we would find for our particle that n would be complex so the "rays of light" trajectories of the particle,can go inside the potential barrier...

[tex]\frac{dS}{dt}+\frac{1}{2m}(\nabla{S})^{2}+V(x)-\frac{i\hbar}{2m}(\nabla^{2}{S}) [/tex]

then we could form the complex potential:[tex]U=V(x)-\frac{i\hbar}{2m}(\nabla^{2}{S})[/tex]

and the Classical equation of Motion in the form:

[tex]m\frac{d^{2}x}{dx^{2}}=-\nabla{U} [/tex]

How do we solve equation (1)?...for example we use a trial function for S=f(r,t) then we calculate [tex]\nabla^{2}{f(r,t)[/tex] and introduce it into equation (1),solve S for this function f(r,t) and again we introduce into the differential equation to find another value of S more accurate than before.

Complex trajectories...are they allowed?..remember that the particle can be into a "classical forbidden" region,then if we use the eikonal equation of Optics [tex](\nabla{S})^{2}=n^{2} [/tex] with n the refraction index of the material we would find for our particle that n would be complex so the "rays of light" trajectories of the particle,can go inside the potential barrier...

Last edited: