Re: operator satisfying relations [H,T] = i hbar
- From: Stephen Parrott <postnews@xxxxxxxxxxxxxxx>
- Date: Fri, 14 Dec 2007 07:00:41 +0000 (UTC)
[snip]
By the way, although one cannot define a Hermitian time operator an
oscillator with frequency w, one can still define a perfectly good
time observable using so-called POVMs (probability operator valued
measures). In particular, denoting the n'th energy eigenstate by |n>,
and the period 2pi/w by T, one defines the kets |t> by
|t> = (1/sqrt{T}) sum_n exp{ -i nwt } |n> .
It is easy to show that
exp{ -i hbar H tau } = | t + tau >,
and that
int_0^T |t> <t| dt = 1.
The probability of the "time" being measured to have the value t in
[0, T) for state |psi> is then given by
p(t|psi) = | <t|psi> |^2 .
One can similarly define POVM time observables for the free particle
and various other (but not all) physical systems (see, eg,
"Probabilistic and Statistical Aspects of Quantum Theory" by A. S.
Holevo).
I think this is very misleading. The sum written to "define" |t> does
not converge, so the reader has to guess how it might be translated into
meaningful mathematics. My guess is as follows.
The Hamiltonian H would presumably satisfy H|n> = n|n>. Since an
unquoted part of the post specified that H be bounded below by 0, the
sum should be over n >= 0.
Taking w := 1 for notational simplicity, the Hilbert space with
orthonormal basis the set of |n> could be realized as the set of
square-integrable functions on [0, 2pi] with Fourier transforms in the
span of functions t -> exp(int) with n >= 0. Such a function would be
denoted |n>, and the action of the Hamiltonian on such a function (to
assure H|n>=|n>) would be to differentiate it (and multiply by i). That
is, the Hamiltonian is what is usually called the "momentum" operator,
except that it operates on square-integrable functions on [0,2pi]
instead of on functions defined on the whole real line.
The operator T would then be multiplication by the coordinate function,
f(t) -> tf(t), so as to assure HT-TH = i. Formally, the (generalized)
eigenfunctions |t> for T would be as the poster stated (essentially,
they are Dirac delta "functions"). Thus the "time" operator" would be
what is usually called the "position" operator.
I think any usual translation of the poster's undefined notation into
meaningful mathematics would be essentially the same as the above, apart
from notation. But if it were correct, it would contradict the result
that HT-TH = i is impossible for H bounded below. What is wrong?
First of all, the alleged operator T does not preserve the stated
Hilbert space. That is, if a function f is a linear combination of |n>,
then Tf need not be, so the mathematics turns out to be nonsense.
It is also probably physical nonsense. Where in the world would one
find a physical system, on which one could do actual experiments, whose
"energy" turns out to be identical to its "momentum", and whose "time"
turns out to be identical to its "position"?
My intention is not to criticize the poster with username "a student".
He probably is a student, and she probably got the above nonsense from
some book or paper. There is a lot of such nonsense in the literature
(check the literature on the "quantum phase" operator).
The point of this post is that one has to read the physics literature
with a great deal of caution, realizing that nonsense may even be more
the rule than the exception. The physical meaning of formal algebraic
calculations should always be carefully examined.
.
- Follow-Ups:
- Re: operator satisfying relations [H,T] = i hbar
- From: a student
- Re: operator satisfying relations [H,T] = i hbar
- References:
- operator satisfying relations [H,T] = i hbar
- From: mingyuming
- Re: operator satisfying relations [H,T] = i hbar
- From: a student
- operator satisfying relations [H,T] = i hbar
- Prev by Date: Parity Calorimetry Experiment, III
- Next by Date: Re: Open poll on "What changes for special and general relativity?"
- Previous by thread: Re: operator satisfying relations [H,T] = i hbar
- Next by thread: Re: operator satisfying relations [H,T] = i hbar
- Index(es):
Relevant Pages
|
|
