Question

The pores in zeolite catalysts are so small that quantum mechanical effects on the distribution of atoms and molecules within them can be significant. Calculate the location in a box of length L at which the probability of a particle being found is 50 per cent of its maximum probability when n =1

Answer 1

For n = 1 the wavefunction of a particle in the box is just half a sine wave, with maximum in the middle and zero at the walls. What you need to know is where, along half a wavelength from 0 to L, a sine wave reaches half of its maximum amplitude. In short, where is the arcsine equal to 1/2?

Fortunately this is one of the easy arcsines, and the answer is 30 degrees. That is, the sine of 30 degrees is 1/2. Let's say your box goes from 0 to L. Then 0 degrees is at 0 and 180 degrees (half the sine wave) is at L. The sine will reach 1/2 its maximum amplitude at 30 degrees, or 30/180 L = L/6, and again (as it declines) at 150/180 L = 5L/6.

Answer 2

Drat, I gave you the wrong answer. On re-reading your question, I realize you asked for where the probability, not the probability amplitude, was half the maximum. Since probability is proportional to the wavefunction squared, what you want is where the sine equals the square root of 1/2, which is 45 degrees, not 30. hence the answers should be L/4 and 3L/4 instead. Sorry!

Answer 3

thank you very much!
even though i am not good at quantum theory, i'll try.