The Double-Slit Experiment*

prepared by

Frank Rioux

Illumination of a double-slit screen with a coherent particle beam leads to a Schrodinger "cat state" that can be represented by a linear superposition (unnormalized) of two Gaussian wavepackets. The probability distribution function in coordinate space, |Y(x)|^{2}, at the slit-screen for this "cat state" is shown below.

The slits localize the particle in the x-direction which leads to a spread in the x-component of the momentum required by the uncertainty principle, DxDp_{x} > h/4p. The momentum wave function is obtained by a Fourier transform of the coordinate-space wave function.

Evaluation of the integral yields,

The momentum probability function in the x-direction is |F(p_{x})|^{2} and simplifies to the expression given below when evaluated.

This momentum probability function is displayed below.

Because the arrival at position *x* on the detection screen is proportional to *p*_{x} it is also proportional to |F(p_{x})|^{2}. In other words, the particle distribution at the detection screen is determined by the momentum distribution at the slit screen. This means the position measurement at the detection screen is effectively a measurement of the *p*_{x}. Therefore, the particle distribution at the detector screen will have the same shape as shown in the figure above.

In summary, the double-slit experiment clearly reveals the three essential steps in a quantum mechanical experiment:

1. State preparation (interaction of the incident beam with the slit-screen)

2. Measurement of an observable (arrival of scattered beam at the detection screen)

3. Calculation of expected results of the measurement step

*The preparation of this tutorial was stimulated by reading "Quantum interference with slits" by Thomas Marcella which appeared in *European Journal of Physics* **23**, 615-621 (2002). This paper offers a lucid and novel quantum mechanical analysis of a very important experiment.

**Additional references:**

R. P. Feynman, R. B. Leighton, and M.Sands, *The Feynman Lectures on Physics*, Volume 3; Addison-Wesley; Reading, 1965, Chapters 1 and 3.

R. P. Feynman, *The Character of Physical Law*; MIT Press: Cambridge, 1967, Chapter 6.

A. Tonomura, J. Endo, T. Matsuda, T. Kawasaki, and H. Exawa, "Demonstration of single-electron buildup of an interference pattern" *Am. J. Phys*. **57**, 117-120 (1989).

D. Leibfried, T. Pfau, and C. Monroe, "Shadows and Mirrors: Reconstructing Quantum States of Atom Motion" *Phys. Today* **51(4)**, 22-28 (1998).

The double-slit experiment with single electrons was recently selected (informally) as physics most beautiful experiment. The following web reference traces the history of double-slit interference experiments from the time of Thomas Young to the present, presenting numerous literature references in the process: **http://physicsweb.org/article/world/15/9/1**.