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24
22
/
2
2
/
2
21 1
21 1
55
( ) 
 (4
)
sin
(4
)
sin
.
16
ra
rr
Pr
r
e
r
e
aa
ψ
πθ
π
θ
−−
±±
⎛⎞
==
=
⎜⎟
64π
⎝⎠
(c) The total probability density for the three states is the sum:
( )
222
2
210
21 1
21 1
210
21 1
21 1
44
/2
2
2
/
()
() 
 
)
11
cos
sin
sin
.
8
PrP rP r
r
ee
ψψ
θθθ
+−
+
−
++=
+
+
⎡⎤
=+
+
=
⎢⎥
8
⎣⎦
The trigonometric identity cos
2
+ sin
2
= 1 is used. We note that the total probability
density does not depend on
or
φ
; it is spherically symmetric.
49. (a)
210
is real. Squaring it, we obtain the probability density:
/
2
2
/
2
210
210
( )

 (4
)
cos
(4
)
cos
.
32
r
e
r
e
=
π8
(b) Each of the other functions is multiplied by its complex conjugate, obtained by
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This note was uploaded on 06/03/2011 for the course PHY 2049 taught by Professor Any during the Spring '08 term at University of Florida.
 Spring '08
 Any
 Physics

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