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#
Michael Weiss: Vacuum energy,
source of all of humanity's future energy needs

OK, time evolution. We have:
Coh*(z) = *exp(-|*z*|^{2}/4)* e*^{za*/
}^{sq2}* *|0>
and

*H = (aa*^{*} + a^{*}a)/2 = a^{*}a + 1/2
including the vacuum energy this time, source of all of humanity's future
energy needs. (Right.)

We want to know how Coh*(z)* evolves (in the Schrödinger picture,
since it *doesn't* evolve in the Heisenberg picture). I.e., we want
to compute

*e*^{-iHt} Coh*(z)*
Now it would be nice if we could just plug in the formulas we just obtained.
Alas, we'd need to compute *e*^{a*a} *e*^{za*}
(give or take an annoying factor), and the Baker-Campbell-Hausdorff formula
we have only helps with *e*^{A}e^{B} when [*A*,*B*]
commutes with *A* and *B*. Is that the case here?

[*a*^{*}a, *a*^{*}] = [*a*^{*},*a*^{*}]*a*
+ *a*^{*}[*a*,*a*^{*}] = 0 + *a*^{*}
= *a*^{*}.
No such luck, the commutator does *not* commute with *a*^{*}a.

Probably we could do something slick anyway, but at least the dull way
is quick.

Coh*(z) = *exp(-|*z*|^{2}/4) SUM_{n}
(z/sq2*)*^{n}/ sqrt(*n*!) |*n*>
*e*^{-iHt} |*n*>* = e*^{-it/2} e^{-int}
|*n*>
so

*e*^{-iHt} Coh*(z) = *exp(-|*z*|^{2}/4)
*e*^{-it/2} SUM_{n} (z/sq2*)*^{n}/
sqrt(*n*!) *e*^{-int} |*n*>
but since *e*^{-int} = (*e*^{-it})^{n},
we can fold this into the *(z/*sq2*)*^{n }factor and
get:

*e*^{-iHt} Coh*(z) = e*^{-it/2} Coh*(ze*^{-it})
So after 2* pi* seconds, we've got minus the original state vector.

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Weiss: Dangerous signs
*Michael Weiss*

*3/10/1998*