Normalization of a wavefunction

EmmaLemming

1. The problem statement, all variables and given/known data

This is a multi-choice question.

A particle of unit mass moving in an infinite square well,

V = 0 for lxl ≤ a
V = ∞ for lxl > a

is described by the wavefunction, u(x) = A sin (3∏x/a)

If the wavefunction is normalised, What is A?

a) 1/2a
b) 1/√2a
c) 1/√a


2. Relevant equations

I know that the integral of the wavefn squared is equal to 1 because it has to exist somewhere but when I tried integrating it, it either all went to 1 or ∞.

I know how to do this question, I just can't. An easy to follow mathematical proof would be most helpful.

3. The attempt at a solution

I am integrating between ∞ and -∞ is that correct?

so far i've got that

∫ A² sin² (3∏x/a) dx = 1

using the identity: cos (2x) = 1 - 2 sin²(x)

= A²/2 ∫ 1 - cos (6∏x/a) dx = 1


And now i'm stuck...

BruceW

You need to be careful here. This is not true for all x. Its in an infinite square well, so what will the wave function be outside of that well?

EmmaLemming

Ah ok, I thought the wave function wouldn't exist outside of the square well so the wave function would be zero..?

I got the sin (3∏x/a) wavefunction given to me in the question so I just took it as true.

BruceW

yes, you're right. The wave function outside the square well is zero. and inside the well, it is sin (3∏x/a). Also, you were right that the integral is from -∞ to ∞. But what is the integrand for lxl > a ?

EmmaLemming

I don't know. This is just a guess would it be,

between 0 and a

∫ A sin (3∏x/a) dx

This is where I get confused, because I thought to do this question all I have to do is square the wavefunction and integrate between -∞ and ∞.

Is there another step before hand?

BruceW

the wave function is A sin (3∏x/a) for lxl ≤ a and it is zero for lxl > a

So you do need to 'square the wavefunction and integrate between -∞ and ∞', but the wave function will be zero for lxl > a

In other words, you need to identify the different 'sections' and integrate each section, to get the integration over all space.

EmmaLemming

Sorry if I'm being slow and thanks for your help.

but would one of the sections be an integral between a and -a ?

Do I do that and then integrate between -∞ and ∞?

BruceW

no worries. Yeah, one section would be from -a to a. The total integral is from -∞ to ∞, so what would the other sections be?