A Quantum Mechanical Interpretation of Single-slit Diffraction

Or, Using Diffration Phenomena to Illustrate the Uncertainty Principle

Frank Rioux
Department of chemistry
College of St. Benedict and St. John's University

Diffraction has a simple quantum mechanical interpretation based on the uncertianty principle. Or we could say diffraction is an excellent way to illustrate the quantum mechanical uncertainty principle.

A screen with a single slit of width, w, is illuminated with a coherent photon or particle beam. The normalized coordinate-space wave function at the slit screen is,

The coordinate-space probability density, |Y(x,w)|², is displayed for a slit of unit width below

Since the slit-screen measures position, it localizes the incident beam in the x-direction. Accoring to the uncertainty principle, because position and momentum are incompatible, or conjugate, observables, this measurement must be accompanied by a delocalization of the x-component of the momentum. To see this it only necessary to Fourier transform Y(x,w) into momentum space to yield the momentum wave function, F(p_x,w).

It is the momentum distribution shown below that is projected onto the detection screen. Thus, a position measurement at the detection screen is also a measure of the x-component of the particle momentum.

In this figure we see the spread in momentum required by the uncertainty principle, plus interference fringes due to the fact that the incident beam can imerge from any where within the slit, allowing for constructive and destructive interference. If the slit width is decreased the position is more precisely known and the uncertainty principle demands a broadening in the momentum distribution as shown below.

Slit width = 0.5

If the slit width is increased the position uncertainty increases and the uncertainty in momentum decreases.

Slit width = 2

As a final example, if the position becomes for all practical purposes certain (infinitesimally thin slit), the momentum distribution becomes completely uncertain. In other words all momentum values have an equal probability of being observed for a quantum particle with a well-defined position. This is shown in the figures below.

The method used here to calculate single-slit diffraction patterns (momentum-space distribution functions) is easily extended to multiple slits, and also to diffraction at two-dimensional masks with a variety of hole geometries.

Primary source: "Quantum interference with slits" by Thomas Marcella which appeared in European Journal of Physics 23, 615-621 (2002).