Functional Derivative

As in path integration one deals with classical functions and functionals rather than with operators then fairly simple classical manipulations of path integration can lead to non-trivial quantum mechanical identities. We consider a simple example of the functional derivative that relates some functional, say, F[q(t) changes by y(t) and the transition amplitude К remains invariant:

and

Here the functional derivative

is also a functional. The action S is also a functional ofy(s).

Suppose we have the function F(q,q,t) then

or

or

But at the end points:

So,

We consider the functional F[ So,

We examine an infinitesimal time slice: then

and

We take the following example then

where the derivative of F(s) is simply given by the Dirac delta distribution:

From

or

then again

Functional Derivative of the Action Functional

& s

We find the functional derivative ——- of the action functional by choosing q(t) as the classical path.

dq(s)

The nearbypaths, which are not chosen by the system, we denote by q(t). The classical path extremizes the action S in comparison with all neighboring paths,

with the same end points q(t_a) = q_a and q(t_b) = q_b. The mathematical problem (calculus of variations) finds the curve for which the action is an extremum (Figure 13.1)

The variation in the action is zero (the action is stationary) near the extremum for all variations of the path about the classical path such that the quantity y(t) vanishes at the end points:

This may be done as or

or

FIGURE 13.1 Illustrates the path of a particle moving from a starting coordinate q(t_a) = q_a to an ending coordinate q,, of a given path.

From the condition in 13.22, then the first summand is zero and

So, the functional derivative of the action functional S is understood as the functional:

The Euler-Lagrange equation of motion for the classical path q(t) is found from the condition that and

We vary now the end point

but keeping (q_a, t„) fixed while using the definition of the canonical momentum p conjugate to q: then

So,

We find now the energy E_b (Energy Function or Hamiltonian) at b:

From here,

or

Equations 13.33 and 13.36 are the so-called Hamilton-Jacobi equations.

Example 1

We consider the example of a free particle with Lagrange function:

The classical action has been found in 5.72 as From the Hamilton-Jacobi equations:

Example 2

We examine the harmonic oscillator where the classical path q(t) and classical action S_d[0,T] have been obtained respectively in 5.121 and 5.125:

From the Hamilton-Jacobi equations: