# Feynman Path Integrals

### From FSUPhysicsWiki

The path integral formulation of quantum mechanics was developed in 1948 by Richard Feynman. The path integral formulation is a description of quantum theory that generalizes the action principle of classical mechanics. It replaces the classical notion of a single, unique trajectory for a system with a sum, or functional integral, over an infinity of possible trajectories to compute a quantum amplitude.

The classical path is the path that minimizes the action.

This formulation has proved crucial to the subsequent development of theoretical physics, since it is apparently symmetric between time and space. Unlike previous methods, the path-integral offers us an easily method by which we may change coordinates between very different canonical descriptions of the same quantum system.

For simplicity, the formalism is developed here in one dimension.

In the path integral formalism, we start by writing the amplitude for a particle at position at time to move to a position at time as a path integral. This path integral is

where is the action for the the path and the integral is defined as

where is a number of "slices" of length that we divide the time axis up into. Essentially, we define the path integral as a limit of an integral over all possible values of the particle's intermediate positions on its path from to

The action is given by the time integral of the Lagrangian, just as in classical mechanics:

where

is the Lagrangian.

Our choice of notation for this path integral, is motivated by the fact that it serves as a "kernel" for an integral giving the wave function in terms of This integral is

## Obtaining the Schrödinger Equation From the Path Integral Formalism

As a justification of this method, we will show that it reproduces the Schrödinger equation. The following derivation follows that of Feynman. Let us begin by assuming that the elapsed time is so small, that we may approximate the path integral with a single "time slice" of that length. In this case, the kernel is just and the action is just its average over the time interval times is length:

The kernel now becomes

so that the wave function is

Now we introduce the variable, so that the integral becomes

We now expand the wave function in the integral in powers of up to second order and the factor involving the potential in powers of up to first order. We also drop all dependence of the potential on The result of this is

The problem has been reduced to the evaluation of Gaussian integrals. Using the formulas,

and

we obtain

We now multiply out the right-hand side and retain only terms that are first order in This gives us

Rearranging, we get

Finally, taking the limit, and renaming to we finally arrive at the familiar Schrödinger equation,

As a final remark, we note that using the Feynman path integral formulation of quantum mechanics is more complex than solving the Schrödinger equation to obtain the dynamics of a quantum particle. Why, then, is this formulation mentioned in textbooks and where it may be useful?

For a single-particle problem, using the Schrödinger equation is definitely easier. However, to study a many-body system, solving the Schrödinger equation can be rather complicated and messy (let's just say sometimes impossible), while the Feynman path integral is a good tool for dealing with many-body problems by writing everything in terms of field operators. More importantly, the generalization of quantum mechanics to relativistic problems can be done in terms of field operators via the Feynman path integral formulation. These applications, while of great interest, are beyond the scope of the present work.