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59.
(a)
ψ
210
is real. Squaring it, we obtain the probability density:

ψ
210

2
=
r
2
32
πa
5
e
−
r/a
cos
2
θ.
Each of the other functions is multiplied by its complex conjugate, obtained by replacing
i
with
−
i
in the function. Since
e
iφ
e
−
iφ
=
e
0
= 1, the result is the square of the function without the
exponential factor:

ψ
21+1

2
=
r
2
64
πa
5
e
−
r/a
sin
2
θ
and

ψ
21
−
1

2
=
r
2
64
πa
5
e
−
r/a
sin
2
θ.
The last two functions lead to the same probability density.
(b) The total probability density for the three states is the sum:
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 Fall '08
 SPRUNGER
 Physics

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