- #1

ja07019

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## Homework Statement

Hello today I am solving a problem where an electron is trapped in a potential well. I have a solved Schrodinger's Equation. I am having problems in figuring out what the wave function should be. When I solved the equation I got a complex exponential. I know I cannot use the complex exponential because when multiplied by its complex conjugate, the answer is simply 1. The book says that the real component is the symmetric solution while the imaginary component is the asymmetric solution. So I tried to calculate the expected value but got an answer that was different from the one given by the textbook. Note that I tried both the symmetric solution and the asymmetric solution. I am asked to calculate the average value of (z-<z>)

^{2}. The potential well is in only 1 dimension and it starts at -L and ends at L.

## Homework Equations

<A> = ∫Ψ*AΨdx

Ψ=Aexp(ikz)

ψ(symmetric)=Acos(kz) and thus ψ* = Acos(kz)

ψ(asymetric) = Asin(kz) and ψ*Asin(kz)

I know that <z> = 0

## The Attempt at a Solution

Here I tried to compute both of the following integrals,

<(z-<z>)

^{2}> = <z

^{2}> = ∫[Azcos(kz)]

^{2}dz

and likewise I also tried to compute

<z

^{2}> = ∫[Azcos(kz)]

^{2}dz.

Both of these are definite integrals. I assumed that they were integrated from -L to L. The answer given in the book is [1-(-1)

^{n}(6/(nπ)

^{2})]*a

^{2}3.

When I attempted the symmetric solution, (the one with the cosine term), I yielded a solution of L

^{3}[(2(nπ)

^{2}+3)/(6(nπ)

^{2})].

The asymmetric solution, which uses sine, is somewhat similar to this solution.