Quantum Mechanics: The Hydrogen Atom
12th April 2010
I. The Hydrogen Atom
In this next section, we will tie together the elements of the last several
sections to arrive at a complete description of the hydrogen atom. This will
culminate in the definition of the hydrogenatom orbitals and associated en
ergies. From these functions, taken as a complete basis, we will be able to
construct approximations to more complex wave functions for more complex
molecules. Thus, the work of the last few lectures has fundamentally been
aimed at establishing a foundation for more complex problems in terms of
exact solutions for smaller, model problems.
II. The Radial Function
We will start by reiterating the Schrodinger equation in 3D spherical coordi
nates as (refer to any standard text to get the transformation from Cartesian
to spherical coordinate reference systems). Here, we have not placed the con
straint of a constant distance separting the masses of the rigid rotor (refer
to last lecture); furthermore, we will keep in the formulation the potential
V
(
r, θ, φ
) for generality.
Thus, in spherical polar coordinates,
ˆ
H
(
r, θ, φ
)
ψ
(
r, θ, φ
) =
Eψ
(
r, θ, φ
)
becomes:
"

¯
h
2
2
μ
1
r
2
∂
∂r
r
2
∂
∂r
+
1
r
2
sinθ
∂
∂θ
sinθ
∂
∂θ
+
1
r
2
sin
2
θ
∂
2
∂φ
2
!
+
V
(
r, θ, φ
)
#
ψ
(
r, θ, φ
) (1)
=
Eψ
(
r, θ, φ
)
(2)
Now, for the hydrogen atom, with one electron found in ”orbits” (note the
quotes!) around the nucleus of charge +1, we can include an electrostatic
potential which is essentially the Coulomb potential between a positive and
negative charge:
1
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V
(

r

) =
V
(
r
) =

Ze
2
4
π
o
r
where
Z
is the nuclear charge (i.e, +1 for the nucleus of a hydrogen atom).
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 Fall '08
 Dybowski,C
 Physical chemistry, Atom, pH, Angular Momentum, Hydrogen atom, dθ sinθ

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