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Unformatted text preview: func0on
to
two
parts.
A
part
that
depends
on
θ
and
a
part
 that
depend
on
ϕ
 Angular
momentum
rota0on
in
the
XY
 plane
 Y (θ , φ ) = Θ (θ ) Φ (φ ) ⎡1 ∂⎛ ∂⎞ 1 ∂2 ⎤ ⎜ sin (θ ) ⎟ + 2 ⎢ ⎥ Y (θ , φ ) = l (l + 1)Y (θ , φ ) sin (θ ) ∂θ ⎝ ∂θ ⎠ sin (θ ) ∂φ 2 ⎦ ⎣ multiplying by sin 2 (θ ) , substitution Θ (θ ) Φ (φ ) for Y (θ , φ ) and dividing by Θ (θ ) Φ (φ ) sin (θ ) ∂ ⎛ ∂⎞ 1 ∂ 2 Φ (φ ) = l (l + 1) ⎜ sin (θ ) ⎟ Θ (θ ) + Θ (θ ) ∂θ ⎝ ∂θ ⎠ Φ (φ ) ∂φ 2 As
before
we
have
a
sum
of
term
that
must
be
a
constant.
One
of
the
term
 depends
only
on
θ,
the
other
depends
only
on
ϕ. SO each of them must be a constant too. We write for the rotation angle ϕ (in the XY plane, along the Z axis)
 1 ∂ 2 Φ (φ ) = − m2 2 Φ (φ ) ∂φ Φ (φ ) = 1 exp ( imφ ) 2π m = 0, ±1, ±2, ... m
is
also
called
the
magne0c
quantum
number.
It
can
be
problem
with
magne0c
 spectroscopy.
 Hydrogen
atom
(con0nua0on)
 ψ ( r,θ , φ ) = R ( r )Y (θ .φ ) 2 1 d ⎛ 2 dψ ⎞ 2 ...
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This document was uploaded on 03/04/2014 for the course CH 354L at University of Texas.

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