The Hydrogen Electron Cloud
In spherical coordinates for threedimensional problems, Schrödinger’s equation becomes
( r
Ψ
r
)
r
/ r
+ (
sin
θ
Ψ
θ
)
θ
/ r
sin
θ
+
Ψ
φφ
/ r
sin
θ
= 2m(
E
– U)
Ψ
/
, where every
incidence of a subscript
r
,
θ
, or
φ
indicates a partial derivative by that variable. For
instance,
Ψ
φφ
means
∂
²
Ψ
/
∂
φ
²
. Obviously, you are not expected to memorize this, or even
fully comprehend it, but it gives a sense of how complicated the problem becomes even
for a simple spherical hydrogen atom where quantum mechanics is involved. The
potential
U
involved here is simply the electric potential energy,
q
/4
π
ε
₀
r
, where
q
is the
charge of the electron. This is a
partial differential equation
that can be broken down
through a process called
separation of variables
into three component equations based on
functions
R
,
Θ
, and
Φ
that are interrelated through constants we will temporarily call
k
₁
and
k
₂
:
( r
R
r
)
r
= – ( 2m
E
r
/
+ mq
r/2
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 Spring '11
 Tibbets
 Atom, Atomic orbital, Partial differential equation, Hydrogen Electron Cloud

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