Hitoshi Kitada (firstname.lastname@example.org)
Tue, 7 Sep 1999 13:40:33 +0900
Dear Matti et al.,
> Dear Hitoshi et all,
> On Tue, 7 Sep 1999, Hitoshi Kitada wrote:
> > Dear Stephen,
> > >
> > > i.e., Psi_E is also an eigen function of H^hat, but with eigenvalue E +
> > > \alpha hbar. Since \alpha is arbitrary, the eigenvalues of H^hat would
> > > take all real values from -\inf. to +\inf, and this is in contradiction
> > > with the existence of a discrete energy spectra."
> > Yes, this is right, insofar as it is considered in the usual framework of
> > I.e. the state vectors are Psi(x), the functions of configuration
> > only.
> > In my case of [time 692], the state vectors are Psi(x,t) =
> > exp(-itH/h)Psi(x,0), the functions of time and coordinates. By this
> > difference, one has
> > i[t, H] = - i[H, t] = i[T, t] = h.
> > Since H is an operator that acts on the functions Psi(x,t) of x and t, and
> > acts as H = -T on them, H naturally has the spectra that is the whole real
> > line, consistent with the argument you quoted.
> > Namely the opeartor H newly defined in [time 692] is different from the
> > Hamiltonian H. The former acts on the four dimensinal space functions
> > (exactly, on Psi(x,t) = exp(-itH/h)Psi(x,0)), while the latter on three
> > dimensional functions. So I reformulated the usual QM to 4 dimesnional
> > then no contradiction and the canonical conjugateness of t to H follows.
> There is perhaps problem with the fact that multiplication by t takes
> you out from the space of energy eigenstates. Or is it a problem?
Yes, this is a problem. E.g., t exp(-itH/h)Psi(x,t), t^2 exp(-itH/h)Psi(x,t),
etc. do not belong to a Hilbert space of 4-dimensional space-time, if such a
Hilbert space could be well-defined at all. I just made an excursion in the
time continent ;-)
> Second problem is that localization in time is not possible due
> to constraints posed by Schrodinger equation: localization would break
> probability conservation. Thus one question whether time operator
> is a useful concept.
I agree. I have the same question. Just made a trip with Stephen's dream :-)
The basic point is that Schrodinger equation
> or any field equation puts constraints on time behaviour and this
> means that one cannot anymore perform arbitrary operations like
> time localization affecting the time behaviour without conflict
> with dynamical law.
You are right again. I completely agree. This is the same problem if it is
possible to construct a four dimensional version of the Hilbert space. What I
proposed is that if the space of states could be thought as the totality of
the QM orbits exp(-itH/h)Psi(x,t), then the conjugateness of t to H is
trivial. This is an identical propsoition by nature of positing the problem.
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