Normalizing the wave function of a free particle

[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!

 

[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}.

 

[maethros]

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}.

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.

 

[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

 

[maethros]

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

Thx, but I know how normalization normally works

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

 

[StatMechGuy]

Okay, so let me ask you what \int_{-\infty}^{\infty} dx e^{\imath (k - k&#039;) x} is. Once you figure that one out, I think you could probably normalize the wave function pretty well.

 

[Meir Achuz]

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.

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.

 

[maethros]

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.

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.

 

[Meir Achuz]

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