Radial Wavefunctions

by Corinne Manogue and Kerry Browne
Copyright 2010 Corinne Manogue

In this worksheet, you will explore the graphs of the radial wavefunctions. 

We will start by setting Zq, the charge on the nucleus, and a0, the Bohr radius, below:

Zq=1
a0=1

We can now define the orbital quantum number, l, and the radial quantum number, n:

n=2
l=0

Before we calculate the radial wavefunction, let's see what the Laguerre factor in it looks like. To do this, first we define the Lagurre polynomials (Laguerre), and the associated Laguerre polynomials (ALaguerre).

Laguerre[q__,ρ__]=Exp[ρ]*D[ρ^q*Exp[-ρ],ρ]
ALaguerre[q__,p__]=D[Laguerre[q,ρ],ρ]
Simplify[ALaguerre[n+l,2*l+1]]

Plot[Simplify[ALaguerre[n+l,2*l+1]],{ρ,0,10*n},PlotStyle->Blue,AxesOrigin->{0,0}]

This is written with the variable ρ, which is the dimensionless parameter we used in class. We need to transform back to obtain the correct dependence on r.

Clear[n,l,a0,Zq]
Lr[n__,l__]=ALaguerre[n+l,2*l+1]/.{ρ->2*Zq*r/(a0*n)}

The normalization is given by,

Nom[n,l]=-(Sqrt[(2*Zq/(a0*n))^3*(n-l-1)!/(2*n*((n+l)!)^3)]*(2*Zq/(n*a0))^l)

Now comes the time to put it all into the radial wavefunction, normalizations, asymptotic values and all:

Rad[n__,l__]=Nom[n,l]*r^l*Exp[-Zq*r/(a0*n)]*Lr[n,l]

n:=1
l:=0
a0:=1
Zq:=1
Plot[Rad[n,l],{r,0,12*n}]

The probability density is given by |(Subscript[R, n, l])((|)^2)

Plot[Rad[n,l]^2,{r,0,12*n}]

For a given value of n and l,  but for all values of m, it turns out that the angular dependence of an energy eigenstate is constant. Therefore, if we want to know the probability of finding the particle in a thin shell of radius r, we can just multiply the probability density above by the area of the shell.

Plot[r^2*Rad[n,l]^2,{r,0,12*n}]

Notice what the extra factors of r do at the origin.

Go back to the top of the worksheet and vary the parameters, l and n, and see what happens.