Questions on Hamilton's Principle and Quantum Mechanics

[tobe]

Hi,


reading Sakurai pages 102-103 (see http://www.scribd.com/doc/3035203/J-J-Sakurai-Modern-Quantum-Mechanics ) I found one thing hard to understand:

If the Phase times Planck's constant equals Hamilton's Principal function in the classical limit (i.e. the action for the physically realized path), why is then the "velocity" (eq. (2.4.23)), respectively the probability flux j directed in the direction of INCREASING Action?
It would be the same in classical mechanics if you identify gradient of S with the momentum vector.

Isn't that in opposition to Hamilton's principle that the action has to be minimized (or at least stationary) on the actual physically realized "path"(I know path is the wrong word for quantum mechanics)?
I guess locally increasing doesn't mean that you cannot still achieve a global minimum somehow (my mathematics background is rather poor, so it would be nice if somebody could help me out here, because I find that hard to imagine).


Cheers,
tobe

 

[eljose79]

Dear tobe,

Thank you for your question and for sharing your thoughts on this topic. I understand your confusion and it is a common question among students studying quantum mechanics. The concept of action in quantum mechanics can be quite tricky to grasp, so let me try to explain it in a simple way.

First, let's take a step back and look at classical mechanics. In classical mechanics, Hamilton's principle states that the action of a system is minimized or stationary on the physically realized path. This means that the path taken by a classical object is the one that minimizes the action, which is a measure of the energy of the system. The equation for action in classical mechanics is S = ∫L dt, where L is the Lagrangian of the system.

Now, in quantum mechanics, we have a similar concept of action, but it is defined slightly differently. The action in quantum mechanics is given by S = ∫p dq, where p is the momentum and q is the position of the particle. This is known as the Hamiltonian action.

Now, let's look at the equation you mentioned in Sakurai, eq. (2.4.23), which states that the "velocity" (or more accurately, the probability flux) is directed in the direction of INCREASING action. This may seem counterintuitive, as in classical mechanics, the velocity is directed in the direction of decreasing action. However, in quantum mechanics, the concept of action is different, and the equation is telling us that the probability flux is directed in the direction of increasing Hamiltonian action.

So, why is this the case? It is because in quantum mechanics, we are dealing with probabilities, not definite trajectories like in classical mechanics. The probability of finding a particle at a certain position is given by the wave function, which is a complex number. The phase of this complex number is related to the action of the system. Therefore, the probability flux, which is the rate of change of the probability, is directed in the direction of increasing action, just like the phase is.

To address your concern about Hamilton's principle, it is still applicable in quantum mechanics. The physically realized path in quantum mechanics is the one that minimizes the action, but in this case, it is the Hamiltonian action, not the Lagrangian action. This is because the Lagrangian is not well-defined in quantum mechanics, and the Hamiltonian is the quantity that is conserved in