Foraxedvalueoflanymvalue

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Unformatted text preview: 1 ∂⎛ ∂ψ ⎞ 1 ∂ 2ψ ⎤ − − sin (θ ) ⎥ ⎜r ⎟+ 2 2 ⎝ dr ⎟ r 2 ⎢ sin (θ ) ∂θ ⎜ ⎠ ⎝ 2 me r dr ∂θ ⎠ sin (θ ) ∂φ 2 ⎦ ⎣ e2 − ψ = Eψ 4πε 0 r The
Schroedinger

Eq.
aVer
mul0plying
by
2mr2
and
dividing
by
Ψ
 ⎤ ⎞ 2 ⎡ d ⎛ 2 dR ⎞ 2 me r 2 ⎛ e2 − r + + E ⎟ R ( r )⎥ ⎢⎜ ⎟ R ( r ) ⎣ dr ⎝ dr ⎠ 2 ⎜ 4πε 0 r ⎝ ⎠ ⎦ 2 ⎡ 1 ∂⎛ ∂Y ⎞ 1 ∂ 2Y ⎤ − ⎢ ⎥=0 ⎜ sin (θ ) ⎟ + 2 Y (θ , φ ) ⎣ sin (θ ) ∂θ ⎝ ∂θ ⎠ sin (θ ) ∂φ 2 ⎦ The
top
term
depends
only
on
r
the
lower
term
depends
only
on
θ
and
ϕ Each of the terms must be a constant (say β)
 Angular
momentum
for
hydrogen
 atom
 Β
constant
below
 − ⎤ ⎞ 2 ⎡ d ⎛ 2 dR ⎞ 2 me r 2 ⎛ e2 2 ⎢ ⎜r ⎟+ 2 ⎜ 4πε r + E ⎟ R ( r ) ⎥ = − β R ( r ) ⎣ dr ⎝ dr ⎠ ⎝ ⎠ 0 ⎦ − 2 ⎡ 1 ∂⎛ ∂Y ⎞ 1 ∂ 2Y ⎤ 2 sin (θ ) ⎟ + 2 ⎢ ⎥= β ⎜ Y (θ , φ ) ⎣ sin (θ ) ∂θ ⎝ ∂θ ⎠ sin (θ ) ∂φ 2 ⎦ Mul0plying
the
lower
equa0on
by
Y(θ,ϕ)sin2(θ)
 sin (θ ) + ∂⎛ ∂Y ⎞ 2 ⎜ sin (θ ) ⎟ + β sin (θ )Y + ⎝ ⎠ ∂θ ∂θ ∂2 Y =0 ∂φ 2 The
lower
equa0on
depends
only
on
ϕ
so
we
can
write
Y(θ,ϕ)=Θ(θ)Φ(ϕ)
 Dividing
the
last
equa0on
by
Θ(θ)Φ(ϕ)
we
have
 Angular
momentum
for
the
hydrogen
 atom
 sin (θ ) d ⎛ dΘ ⎞ 1 d 2Φ 2 =0 ⎜ sin (θ ) ⎟ + β sin (θ ) + Θ (θ ) dθ ⎝ dθ ⎠ Φ (φ ) dφ 2 There
are
two
terms,
one
depends
only
on
θ
and
another
that
depends
only
 on
...
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