- #1
chrisd
- 5
- 0
I have read similar threads about this problem but I wasn't able to make progress using them.
Consider an infinite square-well potential of width a, but with the coordinate system shifted so that the infinite potential barriers lie at x=[itex]\frac{-a}{2}[/itex] and x=[itex]\frac{a}{2}[/itex].
Solve the Schrodinger equation for this case to calculate the normalized wave function [itex]\psi[/itex]n(x) and the corresponding energies En
Time Independent Schrodinger Equation (V(x)=0 within the well)
-\hbar2[itex]\frac{1}{2m}[/itex] ([itex]\frac{∂}{∂x}[/itex])2[itex]\psi[/itex]=E[itex]\psi[/itex]
Skipping ahead to the general solution for [itex]\psi[/itex](x) , I get:
[itex]\psi[/itex](x) = Aeikx + Be-ikx, k = [itex]\frac{\sqrt{2mE}}{\hbar}[/itex]
Using the boundary conditions,
[itex]\psi[/itex]([itex]\frac{a}{2}[/itex])=Aeika/2 + Be-ika/2=0
[itex]\psi[/itex]([itex]\frac{-a}{2}[/itex])=Ae-ika/2 + Beika/2=0,
together with some substitution I am able to prove that,
k =[itex]\frac{\pi}{a}[/itex]n
En =([itex]\frac{n\pi\hbar}{a}[/itex])2[itex]\frac{1}{2m}[/itex]
which I believe to be correct for infinite square well.
I run into trouble trying to solve for the constants A and B. My approach was to try and normalize the wavefunction and then use substitution to solve for A or B, but I can't seem to get anywhere after a certain point.
I would expect to get cosines for n=1,3,5... and sines for n=2,4,6... based on what I know about the shape of the wavefunction, but I am unsure how to prove this mathemetically.
Any tips would be appreciated.
Homework Statement
Consider an infinite square-well potential of width a, but with the coordinate system shifted so that the infinite potential barriers lie at x=[itex]\frac{-a}{2}[/itex] and x=[itex]\frac{a}{2}[/itex].
Solve the Schrodinger equation for this case to calculate the normalized wave function [itex]\psi[/itex]n(x) and the corresponding energies En
Homework Equations
Time Independent Schrodinger Equation (V(x)=0 within the well)
-\hbar2[itex]\frac{1}{2m}[/itex] ([itex]\frac{∂}{∂x}[/itex])2[itex]\psi[/itex]=E[itex]\psi[/itex]
The Attempt at a Solution
Skipping ahead to the general solution for [itex]\psi[/itex](x) , I get:
[itex]\psi[/itex](x) = Aeikx + Be-ikx, k = [itex]\frac{\sqrt{2mE}}{\hbar}[/itex]
Using the boundary conditions,
[itex]\psi[/itex]([itex]\frac{a}{2}[/itex])=Aeika/2 + Be-ika/2=0
[itex]\psi[/itex]([itex]\frac{-a}{2}[/itex])=Ae-ika/2 + Beika/2=0,
together with some substitution I am able to prove that,
k =[itex]\frac{\pi}{a}[/itex]n
En =([itex]\frac{n\pi\hbar}{a}[/itex])2[itex]\frac{1}{2m}[/itex]
which I believe to be correct for infinite square well.
I run into trouble trying to solve for the constants A and B. My approach was to try and normalize the wavefunction and then use substitution to solve for A or B, but I can't seem to get anywhere after a certain point.
I would expect to get cosines for n=1,3,5... and sines for n=2,4,6... based on what I know about the shape of the wavefunction, but I am unsure how to prove this mathemetically.
Any tips would be appreciated.
Last edited: