Apartthatdependsonandapart thatdependon

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Unformatted text preview: to
hydrogen
atom
 The
equa0on
below
suggests
that
we
can
write
the
wavefunc0on
as
a
 product.
We
see
independent
part.
A
part
that
depends
only
on
the
disance
 and
a
par
that
depends
on
the
orienta0on
 1 ∂⎛ ∂⎞ 1 ∂2 ⎤ ⎛ ∂⎛ 2 ∂⎞ ⎞ 2⎡ − ⎜ ⎜ r ψ ( r,θ , φ )⎟ − ⎢ ⎜ sin (θ ) ⎟ + 2 ⎥ ψ ( r,θ , φ ) ⎠ ⎝ ∂r ⎝ ∂r ⎟ ⎠ sin (θ ) ∂θ ⎝ ∂θ ⎠ sin (θ ) ∂φ 2 ⎦ ⎣ 2 ⎡ e2 ⎤ −2 me r ⎢ + E ⎥ ψ ( r,θ , φ ) = 0 ⎣ 4πε 0 r ⎦ 2 And
the
sugges0on
 ψ ( r,θ , φ ) = R ( r )Y (θ , φ ) = R ( r ) Θ (θ ) Φ (φ ) Dividing
by
the
wavefunc0on
we
have
 Angular
momentum:
Hydrogen
atom
 2 ⎛ ∂ ⎛ 2 ∂ ⎞ 2 ⎡ 1 ∂⎛ ∂⎞ 1 ∂2 ⎤ ⎞ − r sin (θ ) ⎟ + 2 ⎟ R ( r )⎟ − ⎥ Y (θ , φ ) ⎝ ⎝⎜ ⎠ Y (θ , φ ) ⎢ sin (θ ) ∂θ ⎜ R ( r ) ⎜ ∂r ⎝ ∂r ⎠ ∂θ ⎠ sin (θ ) ∂φ 2 ⎦ ⎣ ⎡ e2 ⎤ −2 me r ⎢ + E⎥ = 0 ⎣ 4πε 0 r ⎦ 2 We
have
a
sum
of
terms.
Two
terms
depending
on
the
distance
and
one
 depends
only
on
the
orienta0on.
For
the
equality
to
hold
each
of
he
term
 must
be
a
constant.
We
already
know
the
solu0on
for
the
angular
part
 ⎡1 ∂⎛ ∂⎞ 1 ∂2 ⎤ ⎜ sin (θ ) ⎟ + 2 ⎢ ⎥ Y (θ , φ ) = l (l + 1)Y (θ , φ ) l = 0,1, 2, ... sin (θ ) ∂θ ⎝ ∂θ ⎠ sin (θ ) ∂φ 2 ⎦ ⎣ We
can
actually
do
more
than
that
since
we
can
also
separate
the
angular
 momentum...
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This document was uploaded on 03/04/2014 for the course CH 354L at University of Texas at Austin.

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