In the Classroom Single-Slit Diffraction and the Uncertainty Principle Frank Rioux College of St. Benedict/St. John’s University, Collegeville, MN 56321; [email protected] Muiño has used single-slit diffraction to provide an in- 2 1 troduction to the uncertainty principle suitable for an un- Φ px, dergraduate physical chemistry course (1). His article 2 provided both a theoretical analysis of single-slit diffraction 2 and a lecture demonstration of the phenomenon using Φ px,1 readily-available equipment. In the current research literature Nairz and colleagues have confirmed the uncertainty prin- ciple in a single-slit diffraction experiment with a beam of C70 molecules (2). And, quite recently, a research team led by Marcus Arndt and Anton Zeilinger performed multi-slit experiments demonstrating wave-particle behavior for −20 −15 −10 −50 5101520 p tetraphenylporphyrin and a fluorinated fullerene, C60F48 (3). x The significance of these results is that tetraphenylporphyrin h is an important biomolecule and C60F48 is the most massive Figure 1. Momentum distribution at the slit screen for slit widths of particle to demonstrate wavelike properties to date. 0.5 and 1.0 a0 (1 a0 = 52.9 pm). The purpose of this short article is to provide an alter- native theoretical analysis of single-slit diffraction based on the Fourier transform between coordinate and momentum space. This approach was recently used by Marcella (4) to The uncertainty principle is clearly revealed—the nar- analyze single- and double-slit diffraction and has been ex- row slit produces a broader momentum distribution. In other tended by the author to more complicated slit geometries (5). words, localization in coordinate space leads to delocaliza- Quantum experiments have three parts: (i) state prepa- tion in momentum space. However, we can also treat this ration; (ii) subsequent measurement of an observable; and effect in a more quantitative manner. (iii) theoretical interpretation of experimental results. In dif- Following Muiño we will assume that the uncertainty fraction experiments, passage through the slit screen repre- in position is the slit width, w. The uncertainty in momen- sents a position measurement that establishes the state of the tum is defined as half the width of the momentum distribu- system in coordinate space. The coordinate-space wave func- tion of the central diffraction band (2). As the momentum tion for a photon or massive particle that has passed a screen distribution is zero for with a slit of width w centered at x = 0 is pwx w = ± n π (3) 0 for x < − 2h 2 it is easy to show that using this criterion the uncertainty in 1 w w h⁄ ͞ Ψ()xw, = for −≤ ≤ x momentum is 2π w. Therefore, the product of the uncer- tainty in position and momentum is greater than h⁄ ͞2 as re- w 22(1) w quired by the uncertainty principle. 0 for x > 2 2πh ∆∆xpx == w 2πh (4) The quantum mechanical interpretation of diffraction is that w the physical property recorded at the detection screen is the The Fourier transform connecting complementary ob- momentum distribution of the diffracted particle. servables is ubiquitous in quantum theory (and, of course, A Fourier transform of Ψ(x, w) into the momentum rep- in our laboratory instruments). Here the transform between resentation yields the momentum-space wave function position and momentum has been used to illuminate the in- w timate relationship between single-slit diffraction and the uncertainty principle. 2 1 i ppxx 1 Φ()pwx , = exp − dx w h h Literature Cited − 2π w 2 1. Muiño, P. L. J. Chem. Educ. 2000, 77, 1025–1027. pw sin x (2) 2. Nairz, O.; Arndt, M.; Zeilinger, A. Phys. Rev. A, 2002, 65, 2h 2h 032109. = 3. Hackermüller, L.; Uttenhaler, S.; Hornberger, K.; Reiger, E.; πw px Brezger, B.; Zeilinger, A.; Arndt, M. Phys. Rev. L. 2003, 91, where px is the x-direction momentum. Thus, according to 090408. quantum mechanics, the diffraction pattern observed is the 4. Marcella, T. V. Eur. J. Phys. 2002, 23, 615–621. Also see: square of the absolute magnitude of the momentum wave French, A. P.; Taylor, E. F. An Introduction to Quantum Physics; 2 function, |Φ(px, w)| . This is shown in Figure 1 for two slit W. W. Norton and Co. Inc.: New York, 1978; pp 331–336. widths. 5. Rioux, F. Eur. J. Phys. 2003, 24, N1–N3 1210 Journal of Chemical Education • Vol. 82 No. 8 August 2005 • www.JCE.DivCHED.org.