How Do You Calculate the Normalization Constant for Radial Wave Functions?

[Khaleesi]

Hi, so I'm having a bit of trouble understanding the normalization of radial waves. I understand that the equation is the integral of ((R^2)r^2)=1 but I'm not understanding how the process works. I need the normalization constant on R32. I got the function to come out to be (((r^2)(Co)(e^(-r/3a))/(27a^3)) so I can take that and plug it into the normalization equation (with r/a=z) to get ((Co)/(27a^3)^2) (a^4) integral of (z^2)(e^(-z))(z^2)dz then I combined both the z^2's so it's now the stuff out front integral of ((z^4)(e^(-z))dz and this is where I'm stuck. From what I keep seeing is that people are getting actual numerical values and I don't understand how that part works. Please help.

 

[Simon Bridge]

The process is the same as when you normalize any wavefunction - the extra r^2 comes from the volume element for spherical-polar coordinates.
You get the numerical values because it is a definite integral. What are the limits of the integration?

 

[Khaleesi]

The process is the same as when you normalize any wavefunction - the extra r^2 comes from the volume element for spherical-polar coordinates.
You get the numerical values because it is a definite integral. What are the limits of the integration?

Well the initial equation states that it's from 0 to infinity. That's were I don't see an actual value coming into place. Unless the bounds somehow change?

 

[Simon Bridge]

Lets makes sure I follow you - you are trying to evaluate $$\int_0^\infty z^4e^{-z}\;dz$$ ... with a bunch of constant terms out the front?
If so - then what do you get for the indefinite integral?

 

[Khaleesi]

Lets makes sure I follow you - you are trying to evaluate $$\int_0^\infty z^4e^{-z}\;dz$$ ... with a bunch of constant terms out the front?
If so - then what do you get for the indefinite integral?

Yes. And I just plugged it in on mathematica and got an answer of 24. I was trying to do it by hand because I hate taking the easy way out, but thanks so much for responding!

 

[Orodruin]

There is a particular trick for solving integrals of this type. Try computing
$$
\int_0^\infty e^{-st} dt
$$
and then differentiate wrt ##s## a few times.

 

[Simon Bridge]

I was trying to do it by hand because I hate taking the easy way out.

... hint: integration by parts.
You need to do the "by parts" trick more than once.