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.

 

[quicknote]

I would say that complex potentials can potentially lead to allowed trajectories, but this concept is still a topic of ongoing research and debate in the scientific community. The use of complex potentials in quantum mechanics is a useful mathematical tool, but it does not necessarily correspond to a physical reality. It is important to note that the classical equation of motion mentioned in the content is a simplified version of the Schrödinger equation and may not accurately describe the behavior of quantum particles.

The idea of complex trajectories is intriguing, but it is important to remember that it is a mathematical concept and not a physical one. The concept of "classical forbidden" regions in quantum mechanics is based on the probability of a particle being found in a certain location, not on its trajectory. Therefore, the use of the eikonal equation of optics in this context may not be applicable.

In order to solve the equation (1), a more rigorous approach would be to use numerical methods or perturbation theory to obtain an approximate solution. The use of a trial function may provide an estimate, but it may not accurately represent the behavior of the particle.

In conclusion, the concept of complex potentials and trajectories in quantum mechanics is a complex and ongoing topic of research. While they may provide useful mathematical tools, their physical interpretation is still a subject of debate. Further research and experimentation are needed to fully understand the implications of complex potentials and trajectories in quantum mechanics.