What,s the meaning of

[eljose]

is there (if any) a physical meaning to the function $$\nabla^{2}(S)$$ ?...we have that $$gra(S)=p$$ p=momentum vector my question is that somehow the operator proposed above is $$\nabla^{2}(S)=div(gra(S))=div(p)$$
although the proposed operator makes no sense in classical Hamilton-Jacobi Mechanics if we make the change of variable $$\Psi=e^{iS/\hbar}$$ inside SE equation you get the Pseudo-HJ equation:

$$dS/dt= (\nabla(S))^{2}+V(x,t)+U_{b}+i\hbar { \nabla^{2}(S) }$$ with m=1/2 and U_{b} is the Bohmian quantum potential.

 

[eljose79]

Thank you for your question about the physical meaning of the function \nabla^{2}(S). The function \nabla^{2}(S) is known as the Laplacian of the action (S) and it has important physical interpretations in classical and quantum mechanics.

In classical mechanics, the Laplacian of the action appears in the Hamilton-Jacobi equation, which is a fundamental equation that describes the dynamics of a system. The Laplacian of the action represents the kinetic energy of the system and is related to the momentum vector through the gradient of the action, as you have correctly mentioned.

In quantum mechanics, the Laplacian of the action also appears in the Schrödinger equation, which is the fundamental equation that describes the dynamics of a quantum system. In this context, it is related to the quantum potential, as you have also mentioned. The quantum potential arises from the quantum potential theory, also known as the de Broglie-Bohm theory, which is an alternative interpretation of quantum mechanics.

In summary, the Laplacian of the action has important physical interpretations in both classical and quantum mechanics. In classical mechanics, it represents the kinetic energy of a system, while in quantum mechanics, it is related to the quantum potential. I hope this helps to clarify the physical meaning of this function.

 

[charng]

The function \nabla^2(S) represents the Laplacian operator applied to the function S. In physics, the Laplacian is often used to describe the second derivative of a function with respect to space or time, and is commonly associated with the concept of curvature. In this case, \nabla^2(S) could be interpreted as the curvature of the function S.

Regarding the physical meaning of this function, it is important to note that the Hamilton-Jacobi equation is a classical equation that describes the motion of a particle in a conservative system. In this context, S represents the action of the particle and \nabla^2(S) could be interpreted as the curvature of the action.

However, when the change of variable \Psi=e^{iS/\hbar} is made in the Schrödinger equation, the resulting equation is known as the Pseudo-Hamilton-Jacobi equation. This equation includes an additional term, i\hbar \nabla^2(S), which is often referred to as the quantum potential. This term arises from the wave-like nature of particles in quantum mechanics and is not present in classical mechanics.

Therefore, while \nabla^2(S) may have a physical interpretation in classical Hamilton-Jacobi mechanics, it takes on a different meaning in the context of quantum mechanics. It can be interpreted as the curvature of the action in classical mechanics, but in quantum mechanics, it represents the quantum potential and plays a crucial role in the behavior of particles at the quantum level.