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, DxDpx > 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(px)|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 px it is also proportional to |F(px)|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 px. 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:
- State preparation (interaction of the incident beam with the slit-screen)
- Measurement of an observable (arrival of scattered beam at the detection screen)
- 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.
- 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.