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Feynman Path Integrals

The Feynman path integral formulation of quantum mechanics is a powerful and elegant way to describe the dynamics of quantum systems. Previously, we have seen the state vector formalism of quantum mechanics, which is based on a few postulates. The path integral formulation is based on the idea that the probability amplitude for a particle to go from one point to another is given by a sum over all possible paths that the particle can take. This is a very different way of thinking about quantum mechanics, and it leads to many interesting results.

Table of Contents​

Electron Diffraction​

We begin with a quantum phenomenon that is very well known: electron diffraction. This is a phenomenon that is often used to demonstrate the wave-particle duality of electrons. In this experiment, we have a beam of electrons that is incident on a double slit. The electrons can either go through the left slit or the right slit, and we can detect them on a screen behind the slits. The screen is divided into many small regions, and we can count the number of electrons that hit each region. The result is a pattern of bright and dark regions, which is characteristic of wave interference. This is a very surprising result, because we would expect that the electrons would behave like particles and go through one slit or the other.

To understand this phenomenon, we need to think about the electrons as waves. The electron passes through both slits, and the two "waves" intefere with each other.

Next, consider what happens if we add another slit. If we add a third slit, the interference pattern becomes more complicated, but it is still there. The procedure is the same—the electron now passes through three slits at once and inteferes with itself.

But now consider what happens if we add a fourth slit, and then a fifth slit, and so on, until we have effectively removed the screen. And then, consider adding a second screen, and then a third screen, and so on. Although the entire double-slit experiment vanishes, the electron still traverses all possible paths and interferes with itself. This is the essence of the Feynman path integral formulation of quantum mechanics.

The Path Integral (Heuristic Derivation)​

We now have a motivation for the idea that particles can take all possible paths. Now we can try to formalize this idea.

Suppose we have a particle that is moving in one dimension. It traverses every possible path from point to point . Let be one such path. Because the particle is a wave, it has a phase that changes with time. It can be visualized as an arrow on a circle, which rotates with time.

If we split the path into small segments , each small segment leads to a small change in the phase . To find this phase, we borrow from wave mechanics; the phase difference is contributed by both the spatial (wavelength) and temporal (frequency) components. The phase difference is given by:

When we sum over all segments, we can write the total phase as:

This is a very complicated expression, but we can simplify it by using the de Broglie relation and the energy relation . This gives us:

Factoring out the , we have:

Now, as , we can write . The sum becomes an integral:

Next, because , we can write:

The term is the kinetic energy, and is the potential energy. Thus, we have the Lagrangian!

Now that we have derived the phase for one path, we can sum over all paths. Recall from wave mechanics that the amplitude of two intefering waves is related to the sum of their phases. In the same way, we can write the total amplitude as a sum over paths:

Because summing over paths is a continuous action, we can replace the sum with an integral:

This is the Feynman path integral, where is the measure of the path integral.

Propagators​

With the informal derivation out of the way, we shall now formally define the path integral. We first consider the matrix elements of the time-evolution operator , which is defined as:

which is to be interpreted as the probability amplitude for a particle to go from point at time to point at time . More specifically, it is a transition amplitude between two states, which is a function of the initial and final states. This is known as the propagator. To see why this is important, consider the position-space wavefunction , which is defined as:

Now we can insert a completeness relation with another set of states :

Hence, the propagator is a kernel that allows us to evolve the wavefunction in time.

To evaluate the propagator, the key is to split the time interval into small intervals of size , where . To that end, beginning with , we can first split the exponential into factors:

Next, we insert complete relations of between each factor of the exponential:

We can use a trick to get rid of the vectors . To do so, we insert complete relations, this time of the momentum eigenstates , between each factor of the exponential, recalling that :

To simplify the notation, we can use a product notation for the integrand:

It is amusing to see this absolute mess of an integral—not only do we have infinitely nested integrals, but the integrand contains infinitely many products, as well as to the power of a matrix.

We shall use a theorem known as the Trotter product formula. In our case, it implies: