130_notes.dvi

16 Hydrogen

The Hydrogen atom consists of an electron bound to a proton by theCoulomb potential.

V(r) =−

e^2

r

We can generalize the potential to a nucleus of chargeZewithout complication of the problem.

V(r) =−

Ze^2

r

Since the potential isspherically symmetric, the problem separates and the solutions will be a
product of a radial wavefunction and one of the spherical harmonics.

ψnℓm(~r) =Rnℓ(r)Yℓm(θ,φ)

We have already studied the spherical harmonics.

Theradial wavefunctionsatisfies the differential equation that depends on the angular momentum
quantum numberℓ,

(

d^2

dr^2

+

2

r

d

dr

)

REℓ(r) +

2 μ

̄h^2

(

E+

Ze^2

r

−

ℓ(ℓ+ 1) ̄h^2

2 μr^2

)

REℓ(r) = 0

whereμis the reduced mass of the nucleus and electron.

μ=

memN

me+mN

The differential equation can be solved (See section 16.3.1) usingtechniques similar to those
used to solve the 1D harmonic oscillatorequation. We find theeigen-energies

E=−

1

2 n^2

Z^2 α^2 μc^2

and theradial wavefunctions

Rnℓ(ρ) =ρℓ

∑∞

k=0

akρke−ρ/^2

where the coefficients of the polynomials can be found from therecursion relation

ak+1=

k+ℓ+ 1−n

(k+ 1)(k+ 2ℓ+ 2)

ak