Unformatted text preview:

Problem 4.9 (Engel)Part A. Are the eigenfunctions of H` for the particle in a one-dimensional box also eigenfunctions of the positionoperator x`?SolutionStrategy. The eigenfunctions of H` are yHxL ="#####2ÅÅÅÅa sin@n p xÅÅÅÅÅÅÅÅÅÅÅaD. If these are also eigenfunctions of x`, they will satisfyx` y=const *y, so let's evaluate the left side of this equation and see what happens.Execution.x`= xx` y=x ybut x is not a constant across the interval 0 Ø a, so different portions of y will be multiplied by different values.Aside. The preceding argument might seem to say that no function can ever be an eigenfunction of the position operator, but this isn't exactly true. Eigenfunctions of x` (call them d or "Dirac delta functions") exist in theory, but they have exceptionally curious (and some might say impossible) shapes.To satisfy the eigenvalue equation, diHxL must equal 0 at all possible x except for one point in space (x = xi). This guarantees that x diHxL is a constant multiple of diHxL at all points in space (think about this). Second, diHxiL =¶, the one point where di does not vanish. This is required so that diHxL is not just a zero function and so that it can be normalized.If we could graph d vs. x, we would see a function that is zero everywhere except for one point where it rises instantly to infinity.Part B. Calculate the average value of x for the case where n = 3 and 5, i.e., yHxL ="#####2ÅÅÅÅa sin@3 p xÅÅÅÅÅÅÅÅÅÅÅaD or "#####2ÅÅÅÅa sin@5 p xÅÅÅÅÅÅÅÅÅÅÅaDSolutionStrategy. The average value of x is also called the expectation value, <x>, and is obtained as the value of the followingintegral:<x> = Ÿ0ay x`y„xNote: this integral relies on these facts: 1) y is a real function, and 2) y has been normalized.Execution.n = 83, 5<·0ax ikjjjjjjj$%%%%%%2a SinAn π xaEy{zzzzzzz2  x83, 5<9a2,a2=The average position of the particle for both the n = 3 and n = 5 states is the center of the box.This is identical to what we expect for a classical particle because a classical particle can be anywhere in the box withequal probability.Comment. The text would like us to generalize this result to the following: the average position of the particle is a/2regardless of n. You might be reluctant to make this generalization, however, because the two examples that werechosen both involved odd values of n. To guarantee the result, we calculate the average value for an arbitrary case of n= m·0ax ikjjjjjjj$%%%%%%2a SinAm π xaEy{zzzzzzz2  x−a H−1 − 2m2π2+ Cos@2mπD + 2mπ Sin@2mπDL4m2π2FullSimplify@%, m ∈


View Full Document

REED CHEMISTRY 333 - Study Notes

Download Study Notes
Our administrator received your request to download this document. We will send you the file to your email shortly.
Loading Unlocking...
Login

Join to view Study Notes and access 3M+ class-specific study document.

or
We will never post anything without your permission.
Don't have an account?
Sign Up

Join to view Study Notes 2 2 and access 3M+ class-specific study document.

or

By creating an account you agree to our Privacy Policy and Terms Of Use

Already a member?