Re: operator satisfying relations [H,T] = i hbar
- From: Stephen Parrott <postnews@xxxxxxxxxxxxxxx>
- Date: Fri, 14 Dec 2007 07:00:44 +0000 (UTC)
[snip]
Then, by standard procedure, if we quantize the above classical
Hamiltonian system (written as P and Q), the classical variable will
become quantum mechanical operators, and the Possion bracket become
commutors. This means, for the oscillator, [P,Q]=ihbar and
[H,Q]=ihbar, where P,Q,H are operators and [,] are commutors. Since P
and Q (and H) are real physical quantities in classical limit, I would
argue they are observables in quantum mechanics for the oscillator. We
can just rename our Q to T is we want to.
I would appreciate if someone can point out the mistakes in the
logical steps.
In essence, you seem to be saying that there exist operators P and Q
satisfying [P,Q] := PQ-QP = i, and that if we rename P as H and call it
a "Hamiltonian" then also [H,Q] = i. And we can rename Q as T and call
it a "time" operator.
Or from a more physical point of view, perhaps you are asserting that
there exists a physical system whose Hamiltonian H is identical to P and
such that if we measure the above Q, then we are really measuring time.
That's all fine, but it misses the point. Your original post seemed to
ask why the above construction does not contradict John Baez's assertion
that for an operator H which is bounded below (as is traditionally
assumed for physical Hamiltonians), there cannot exist an operator T
with [H,T] = 1.
THE ANSWER: The reason there is no contradiction between your
construction and Baez's assertion is that your "Hamiltonian" is not
bounded below.
It is possible to arrange the formalism so that it may APPEAR that your
Hamiltonian may be bounded below, but this is an illusion. If you try
to write out the mathematics rigorously, you will find that it cannot be
done. Trying to force your Hamiltonian to be bounded below causes a
contradiction to appear elsewhere in the argument. A reply to "a
student" (another poster in this thread) explains this more fully.
.
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