Let's consider the Hamiltonian of the hydrogen atom (all constants are 1):
H=2mp2+r1
In coordinate representation of this operators looks like:
p=−i∇r1=(x12+x22+x32)1
Schrodinger equation in this case will be:
H∣Ψ>=E∣Ψ>(−2mΔ+(x12+x22+x32)1)Ψ(x)=EΨ(x)
We will search the solution of the one, by assuming such decomposition:
Ψ(x)=R(r)Y(θ,ϕ)
The Laplasian could be represented in such way:
Δ=r1∂r∂(r1∂r∂)+r21Δθ,ϕ
Taking into account this facts we can write the new equation:
−R1r1∂r∂(r1∂r∂R)−Y1r21Δθ,ϕ(Y)+r1=E(−Rr∂r∂(r1∂r∂R)+r−Er2)−Y1Δθ,ϕ(Y)=0
The solution of the problem:
Δθ,ϕ(Yl,m)=l(l+1)Yl,m
So, there is Legendre polynoms: Yl,m=Pl,m(cos(θ)) . This solution depends from ϕ also.
It's spherical harmoncs for sphere for r=1 . Now, the previous equation have taken the view:
(−Rr∂r∂(r1∂r∂R)+r−Er2−l(l+1))=0
This is equation for Lauguerre polynomials.
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