- #1

WrongMan

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## Homework Statement

An electron coming from the left encounters/is trapped the following potential:

-a<x<0; V=0

0<x<a; V=V

_{0}

infinity elsewhere

the electron has energy V

_{0}

a)Write out the wave function

b)normalize th wave function

## Homework Equations

## The Attempt at a Solution

for -a<x<0

$$Ψ(x)=Acos(kx)+Bsin(kx)$$

$$k^2=\frac{2mV_0}{ħ^2}$$

and for 0<x<a

$$Ψ(x)=Cx+D$$

and 0 elsewhere

i used the sine and cosine because it seemed it would be better for continuity condition in x=0, if you would use exponential form please do explain why.

so this is what my teacher expects for a).

for b)

applying continuity conditions on x=0 i get:

A=D

B=C

and so:$$\int_{-a}^{0}|Ψ(x)|^2=1$$

im a bit confused here, is this the norm or the module? i think its the norm and if so ot might have been worth it to write the wave function in exponential form, so before i transcribe this big integral please clarify this for me.

Furthermore this should look like a particle traped in a box correct? i don't really understand what happens when E=V, i understand the probabiity part, it decays linearly further inside the step, correct?

And what about if E>V

_{0}is it a particle traped in a box, but in the 0-a area the amplitude decreses? And the allowed energy levels for that area start at V

_{0}? what about penetration? and when E is smaller what happens?

Thank you!

Edit:would it be easier if i shifted the potential by -a so that it is in the range [0;2a]?

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