4.3. EXPECTATION VALUE AND STANDARD DEVIATION
49
What are the expectation values of energy, square angular momentum, and
z
angular momen
tum for this state?
Answer:
Note that the square coefficients of the eigenfunctions
ψ
211
and
ψ
21
−
1
are each
1
2
,
so each has a probability
1
2
in the 2p
x
state.
Eigenfunction
ψ
211
has an energy eigenvalue
E
2
, and so does
ψ
21
−
1
, so the expectation value
of energy in the 2p
x
state is
(
E
)
=
1
2
E
2
+
1
2
E
2
=
E
2
=
−
4
.
3 eV.
This is as expected since the only value that can be measured in this state is
E
2
.
Similarly, eigenfunction
ψ
211
has a square angular momentum eigenvalue 2¯
h
2
, and so does
ψ
21
−
1
, so the expectation value of square angular momentum in the 2p
x
state is that value,
(
L
2
)
=
1
2
2¯
h
2
+
1
2
2¯
h
2
= 2¯
h
2
.
Eigenfunction
ψ
211
has a
z
angular momentum eigenvalue ¯
h
, and
ψ
21
−
1
has
−
¯
h
, so the expec
tation value of
z
angular momentum in the 2p
x
state is
(
L
z
)
=
1
2
¯
h
−
1
2
¯
h
= 0
Measurements in which the
z
angular momentum is found to be ¯
h
average out against those
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 Fall '11
 GARVIN
 Standard Deviation, Variance, Angular Momentum, Energy, Momentum, Eigenvalue, eigenvector and eigenspace, Fundamental physics concepts, square angular momentum

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