The Energy States and Radial Wavefunction R(r)
of the Hydrogen Atom

The start point of our talk is our ODE for R(r) that we have obtained before the derivation of the squared kinetic moment eigenfunction Yl,m(J,j):

[−h²/2µ 1/r² ∂/∂r(r² ∂/∂r) + V(r) + h²l(l+1)/2µr²− E] R(r) = 0

One can write down the following V(r) = −Ze²/4peo1/r for the potential of Z-electrons revolving around nucleus. Multiplication by −2µ/h gives finally:

d²/dr² R + 2/r dR/dr + [2µE/h² + 2µ/h²Ze²/4peo1/r−l(l+1)/r²] R = 0

To shorten our last equation we introduce few abbreviations and new radial wavefunction which is assigned by a prime (R' ≡d R/dr ):

ρ = 2 ε r B = Ze²µ/4peoh²ε ε² = −2µE/h² (binding energy)

R'' + 2/ρ R' − [¼−B/ρ + l(l+1)/r ²] R = 0

For high distances (ρ →∞) there are no 1/r ,1/r ² -terms:

R'' − R/4≈ 0 → R(ρ) = C e−ρ/2 or R(ρ) = C e+ρ/2

The second solution is given up since R(ρ) isn't limited for high distances and would be infinite.
The more detailed solution is given elsewhere but we will write here only its short form with the solution obtained:

R(ρ) = e−ρ/2.ρl Σk=0 ak ρk

If we have a look at the ODE and compare coefficients at ρk (ODE should be true for any r and hence for any ρk) we will obtain:

ak+1/ak = l + k + 1 −B/(k + 1) (k + 2l + 2)

The series begins with k = 0. They should finish somewhere in order to stop R-increasing, i.e. one of the coefficients should be 0:

l + k + 1 | − B = 0
½¾¯¾½ |
n |

l, k, 1 are integer numbers, i.e. n = l + k + 1 should be also integer

n = 1, 2, 3, 4 ... 0 ≤ l ≤ n −1 | where l, k = 0, 1, 2, 3.... n is the main quantum number

Introducing B = n we can shorten our equation:

En = −µ/2(Ze²/4pe o h)².1/n²

So the energy is quantized according to the main quantum numbers n and the possible values of kinetic moment are l = 0, 1, 2, ..., n−1 and there are (2l+1) m-values for each l.

For n = 1: l = 0, k = 0 → R(ρ) = C e−ρ/2

or after introduction of r = (2Z/aon) r : R(r) = C e−Zr/nao

with the following abbreviation (Bohr radius) ao = 4peoh²/µe²

The polynomial S akρk in the obtained solution R(ρ) = e−ρ/2.ρl Σk=0 akρk is also referred to as associated Laquere polynomial (Σk=0 akρk ≡ Ln-l-12l+1(ρ)). There is awful mess with the Laguere polynomials because they are defined in another way in some textbooks. We are keeping to mathematical form here which is also applied in the "Mathematica" toolbox. And one can calculate any associated Laguerre polynomial in this way (Rodrigues representation):

Lpq(x) = ex x-q/p!. dp/dxp (e-xxp+q)

It's very simple for any Mathematica user. The following program string gives you back first five associated Laguerre polynomials:
Do[Print[ "n=",n," l=",l," " , LaguerreL[n-l-1,2l+1,x]],{n,1,5},{l,0,n-1}]
The table of polynomials for low level electrons shows they are really simple functions:

Electron | n | l | Ln-1-12l+1 (x)
1 s 2 p 3 d 4 f | 1 2 3 4 | 0 1 2 3 | 1 1 1 1
2 s 3 p 4 d | 2 3 4 | 0 1 2 | 2 - x 4 - x 6 - x
3 s 4 p | 3 4 | 0 1 | 6 - 6x + x² 20 - 10x + x²
4 s | 4 | 0 | 24 - 36x + 12x² - x³

The total solution for the radial part of wavefunction is then as follows

Rn,l = cn,l ·ρl. e−ρ/2 Ln-1-12l+1(ρ)

where the constant cn,l is also the normalization factor. It can be obtained from the normalization condition:

oò¥ Rn,l · Rn,l · r² dr = 1

If we now substitute ρ with r using r = (2Z/aon) r then we will obtain the following total solution:

Rn,l = -{4(n−l−1)! Z³/[(n+l)!]n4a o ³}½ (2Zr/nao)l e−Zr/nao Ln-l-12l+1(2Zr/nao)

where

ao = 2 r Z/n ρ = h²(4peo)/µe²

The amount of equations for wavefunction radial part almost corresponds to the given in textbooks. You can hardly find a textbook where there is one equation written correctly at least. If you want to calculate everything correctly you can use "Mathematica" toolbox where the distance r is measured in ao units (for instance, r=2 corresponds to 2ao = 0,112nm) :
Rnl[n_,l_]:= 2*Sqrt[z^3/n^4*(n-l-1)!/(n+l)!]*(2*Z*r/n)^l*Exp[-Z*r/n]* LaguerreL[n-l-1,2l+1,2*Z*r/n]
nmax=5; Do[Print[ "n=",n," l=",l," " ,Rnl[n,l]],{n,1,nmax},{l,0,n-1}]

The wavefunctions Rn,l(r) and probability to detect electron on the distance r from nucleus Rn,l2r2 are shown on the next two figures:

Radial wavefunction of hydrogen atom for n = 1, 2 and 3. The ordinate is always [Rn,l(r) m−3/2]·10−8.

Radial detection probability in the hydrogen atom for n = 1, 2, and 3. The ordinate is always [e² Rn,l²(r) · 10−15.

The obtained images of all wavefunctions y(r,J,j) = R(r) .Y(J,j) for hydrogen-like atoms are given in the next chapter.