On a Relation between the Heisenberg and deBroglie Principles Oliver G. Ludwig Villanova University, Villanova PA 19085
Most general chemistry texts have a brief discussion of the duality of both light and matter, in which they mention the Heisenberg uncertainty principle and the deBroglie equation for the wavelength of a particle. Seldom, however, does a text do much more with either relation beyond offering some wordy argument about the impossibility of locatine articles of verv small mass and the irrelevance of both p&xiples to ma~oscopicparticles. This paper suggests a way of looking a t wave-particle duality with a more concrete application of both relations and a more insightful internretation of the deBroelie wavelendh. ~ ehzt be the uncertainti in position&d A(mu) = m.Au be that in momentum. Then the Heisenberg uncertainty relation in its simplest form gives hr.mAu 2 hl4x
or, isolating the position uncertainty:
Now suppose that we know the speed to give or take 5%, so that u is known to be between u - 0 . 0 5 ~and u + 0.05u, or that Au = 0.10~.In general, if we know the speed to within ff 100a)%,where a is some fraction, then u is known to be in the interval (u - uu,u + au) and Au is 2a w. We use this in eq 1to obtain hr 2 hX4mnuC2a) Now using the deBroglie equation h = hlmu we obtain the relation Thus, for a speed uncertainty of f4%, a = 0.04 and hz 2 h. This says that only if the interval of uncertainty of speed is as wide as 8%can the uncertainty ofposition be as small
28
Journal of Chemical Education
as the deBmglie wauelength! For a knowledge of speed better than this, Ax must exceed the deBroglie wavelength by a still greater amounefor the case of give or take 1% in knowledge of the speed, Au = 0.020, the position is uncertain by (at least) 1/25 x 0.01 or four deBroglie wavelengths. The "take home" message for the student is that the deBroglie wavelength measures the best we can talk about as to position when we have even a rough estimate of the speed of the particle. The accompanying table should give students a feeling for how well we can know the position of particle when we know the speed to one significant digit. (i.e., the speed is known to within f5% so a = 0.05). To facilitate comparisons, we choose as a typical chemical energy the O-H bond energy, 4.8 eV, and express the minimum position uncertainty in units of the O-H bond length, 0.958.L With these data, eq 2 produces the results in the third column of the table. Clearly the electron cannot be located even within the size of several water molecules, the proton is somewhat fuzzy too, but a strikeout in baseball should not be blamed on the uncertainty principle. The author would like to thank Professor Saul Shupaek for many interesting discussions and Coach Larry Shaue for the Rule Book's weight of a baseball: "between 5 and 51/4 ounces". Particle
bond lengths
Axmi&-H
4.8eV electron
5.6
4.7
4.8eV proton
0.13
0.11
100 mph, 1459 baseball
1 x 1o - ~ ~
8x
loa5
