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:
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(Ze²/4peo |
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.ρlΣk=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):
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) |
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)!]n4ao³}½ (2Zr/nao)l e−Zr/nao Ln-l-12l+1(2Zr/nao) |
where
ao = 2 r Z/n
ρ
= |
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.