Normalizing the wave function of a free particle

1. Apr 19, 2007

maethros

Hello!

Can somebody tell me, how it is possible to normalize the wave function of a free particle using the Dirac delta function?

Thanks!

2. Apr 19, 2007

Meir Achuz

There are two methods that are commonly used:
1. Box normalization. Space is assumed to be contained in an LXLXL box.
After calculating, say, a scattering amplitude, taking the limit L-->\infty
gives a ifntie result if done carefully.

2. Delta function normalization <x|x'>=\delta(x-x')/(2\pi)^{3/2}.

3. Apr 24, 2007

maethros

How can i use the 2nd one this in this case? I have the wave function: psi(x) = A*e^ikx + B*e^-ikx with k = sqrt(2mE/h^2).
I think I can take A = 1, but then i don't know how to continue.

4. Apr 24, 2007

valtorEN

normailization is simple.
u have the wavefunction, all u do is square it and integrate, setting equal to 1
so in ur case, int[-inf to inf] A*e^ikx=A^2*e^2ikx=1
pull A^2 from the integral to get A^2 int[-inf to inf]e^2*ikxdx=1 for the first
1/A^2

5. Apr 24, 2007

maethros

Thx, but I know how normalization normally works

But not in this case: Free Particle and I HAVE TO use the DELTA FUNCTION.

6. Apr 24, 2007

StatMechGuy

Okay, so let me ask you what $$\int_{-\infty}^{\infty} dx e^{\imath (k - k') x}$$ is. Once you figure that one out, I think you could probably normalize the wave function pretty well.

7. Apr 24, 2007

Meir Achuz

What are you going to do with the wave function. If you are going to calculate reflection and transmission coefficients, you odn't have to normalize it.

8. Apr 24, 2007

maethros

I only want to know how I can normalize it using the Dirac delta function. That is all.
I never said that i want to calculate the reflection or transmission coefficient.

9. Apr 25, 2007

Meir Achuz

Your \int |psi|^2 will have four terms. Four each term use
\int exp{ikx-ik'x}=(2pi)^3\delta(k-k').

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