Normalization of a wavefunction

[EmmaLemming]

Homework Statement

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

Homework 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.

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]

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

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?