Can complex potentials lead to allowed trajectories?

[eljose]

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

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

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

and the Classical equation of Motion in the form:

$$m\frac{d^{2}x}{dx^{2}}=-\nabla{U}$$

How do we solve equation (1)?...for example we use a trial function for S=f(r,t) then we calculate $$\nabla^{2}{f(r,t)$$ 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 $$(\nabla{S})^{2}=n^{2}$$ 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...

 

[eljose]

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

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

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

and the Classical equation of Motion in the form:

$$m\frac{d^{2}r}{dr^{2}}=-\nabla{U}$$

How do we solve equation (1)?...for example we use a trial function for S=f(r,t) then we calculate $$\nabla^{2}{f(r,t)$$ 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 $$(\nabla{S})^{2}=n^{2}$$ 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...

Remember that Uncertainty Principle says that we can not meassure the variable of postion and momentum because if we meassured the position using photons these photons interacting with the particle would modify its momentum but this does not imply that trajectories won,t exist.