Why Does Normalizing ψ(x,t) Result in Ae^(-2λ|x|)?

[Gemini_Cricket]

Homework Statement

ψ(x,t) = Ae^(-λ|x|)e^(-iωt)

This is a rather long problem so I won't get into the details. I understand how to normalize, and most of the rest of the problem. I also have the solutions manual. I just need an explanation of why this goes to Ae^(-2λ|x|). I can't figure it out.

Homework Equations

I can't think of any that make sense to use.

The Attempt at a Solution

I believe it is because you can add the powers of exponentials, such that e^(x)e^(x) = e^(2x). I do not understand how you can just get rid of the imaginary, angular frequency, or time parts...

Any explanation would be great.

 

[Dick]

Homework Statement

ψ(x,t) = Ae^(-λ|x|)e^(-iωt)

This is a rather long problem so I won't get into the details. I understand how to normalize, and most of the rest of the problem. I also have the solutions manual. I just need an explanation of why this goes to Ae^(-2λ|x|). I can't figure it out.

Homework Equations

I can't think of any that make sense to use.

The Attempt at a Solution

I believe it is because you can add the powers of exponentials, such that e^(x)e^(x) = e^(2x). I do not understand how you can just get rid of the imaginary, angular frequency, or time parts...

Any explanation would be great.

It's because to normalize you need to integrate ψψ*, the wave function times its complex conjugate. The complex conjugate of e^(-iωt) is e^(iωt). e^(-iωt)*e^(iωt)=e^0=1.

 

[Gemini_Cricket]

Ah okay. I had thought it might have something to do with the complex conjugate.

So for the complex conjugate you just get...

e^(-λ|x|)*e^(-λ|x|) = e^(-2λ|x|)

 

[Dick]

Ah okay. I had thought it might have something to do with the complex conjugate.

So for the complex conjugate you just get...

e^(-λ|x|)*e^(-λ|x|) = e^(-2λ|x|)

Sure.