x2
– Expectation
Physics
World
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The
expectation <x2>,
of the Gaussian state
is
defined as
The A in
Eq. (1) is the magnitude of Y.
It’s chosen such that the state is normalized, i.e.,
The
integral in Eq. (4) has the value
This
insures that |Y|2
is a probability density. We can assume A is a real number since a complex value
will differs only by a phase factor which does not change the value of the
expectation. This implies that A has the value
The
probability density for the wave function in Eq. (1) is
The
integral in Eq. (2) then becomes
Make the
following substitution:
Þ x
= a(h + h0),
dx = adh,
The second integral in Eq. (9) vanishes since it’s the integral of an odd function over symmetric limits. The third integral is has the value given in Eq. (5). All that is left to do is to evaluate the first integral.
Change
variables
Eq (11)
can be integrated by parts as follows: Let
To place
this in a more recognizable form we change variables once again:
,
Upon
simplification we get the final result
