# Stuck on a 'normalisation' step

1. May 29, 2009

### bumclouds

Hey guys,

1. The problem statement, all variables and given/known data
I have an assignment, which is to Solve Schrodinger's equations, for a certain potential distribution, which can be divided up into three regions.

A solution for one of the regions is of the form: Ae$$^{kx}$$

If you substitute this into Schrodinger's equation (time independant, one dimension) and solve for k, you get this:

Schrodingers:
$$\frac{-h}{2m} \frac{d^{2}}{dx^{2}} \Psi (x) + Vo \Psi (x) = E \Psi (x)$$

Solve for k:
k = $$\frac{\sqrt{2mE}}{h}$$

I know this part is right because I've seen it written on the board a couple of times, and it's also what I get on paper.

But then there's the next bit, which I don't get. Apparently it's 'normalising k' which I just don't get..

k = $$\frac{\sqrt{2mE}}{h}$$

$$k\overline{^}\overline{}$$ = $$\frac{\sqrt{2mE}}{h}.\frac{L}{2}$$

[E] = $$\frac{\pi^{2}h^{2}}{2ML^{2}}$$

k_hat = $$\frac{L}{2} . \frac{\sqrt{2m}}{h} . \sqrt{E}$$

k_hat = $$\frac{L}{2} . \frac{\sqrt{2m}}{h} . \sqrt{\frac{E[E]}{[E]}}$$

k_hat = $$\frac{L}{2} . \frac{\sqrt{2m}}{h} . \sqrt{[E]} \sqrt{Ehat}$$

k_hat = $$\frac{\pi}{2} \sqrt{Ehat}$$
2. Relevant equations

None.

3. The attempt at a solution

If I rearrange k = $$\frac{\sqrt{2mE}}{h}$$ and make E the subject,

I get ..

E = $$\frac{h^{2}khat^{2}}{2m}$$

and maybe this is where I go wrong.. because I assume E = [E] ?

Subtituting [E] into the second last step in section 2 above yields:

$$khat = \frac{L}{2} khat \sqrt{Ehat}$$

and that doesn't equal the last step >_<

EDIT: ahh.. if I use their definition of [E], [E] = $$\frac{\pi^{2}h^{2}}{2ML^{2}}$$
I arrive at the right answer..

so how did they come up with [E] = $$\frac{\pi^{2}h^{2}}{2ML^{2}}$$??

Last edited: May 29, 2009
2. May 29, 2009

### Cyosis

What is k_hat and what are you trying to achieve, an expression for E?

3. May 29, 2009

### bumclouds

Sorry!! I've edited my above post.. Now it makes more sense :)

What I'm really after is how did they (my teacher) come up with [E] and what is it?

k_hat is apparently a normalised k