Classical Limit of Quantum Mechanics recovered from the Path Integral Formalism

Question

From Zee's Quantum Theory in a Nutshell he explains how the classical limit of quantum mechanics can be recovered from the path integral formalism.

It can be shown that the path integral formalism is:

$$ \langle q_F | e^{-iHT} | q_I \rangle = \int Dq(t)e^{i \int_0^T dt L(\dot{q},q)}$$

He then restores Planck's constant to get

$$\langle q_F | e^{-i/\hbar HT} | q_I \rangle = \int Dq(t)e^{(i/\hbar) \int_0^T dt L(\dot{q},q)}$$

I do not understand what he means by restoring Planck's constant, why he wants to do this and why he is allowed to do this.

Then he says to take the limit $\hbar \rightarrow 0$.

What is the motivation for taking this limit?? I just cannot see what the intuition for doing this is.

Moreover, once you have done this they obtain $e^{(i/\hbar) \int_0^T dt L(\dot{q_c},q_c)}$ where $q_c(t)$ is the classical path.

What does he mean by the classical path? I really dont remember this in my course on Quantum Mechanics so I must be missing something here.

Answer 1

1) I do not understand what he means by restoring Planck's constant, why he wants to do this and why he is allowed to do this.

The Planck's constant should be there from the outset of the calculations to the end, however, in the natural units, it is one. He wants to do this to take the limit where the quantum description is not necessary and the classical description of the system is valid. He is allowed to do so, because as I said the Plnack's constant was already there, in the beginning, we just set it to one.

2) What is the motivation for taking this limit?? I just cannot see what the intuition for doing this is.

As I explained above it is the limit where the quantum system becomes a classical system. The intuition is that if you say that the quantum mechanical constant is zero you should restore the classical description of the system.

3) What does he mean by the classical path? I really dont remember this in my course on Quantum Mechanics so I must be missing something here.

The classical path is the one that extremizes the action. You should recall the courses on Euler-Lagrange formalism. In my university, we learned this is the Classical Mechanics course.

Punchline: Start from a quantum system, restore the quantum mechanical constant, take the limit that it goes to zero to see if you can restore the classical dynamics as you should. The classical path is related to the Euler-Lagrange equation or the extremization of the action if you prefer.

Cheers!!!