Matti Pitkanen (matpitka@pcu.helsinki.fi)
Tue, 7 Sep 1999 07:00:44 +0300 (EET DST)
Hi all,
some comments about time operator below.
On Mon, 6 Sep 1999, Stephen P. King wrote:
> Dear Hitoshi et al,
>
> Hitoshi Kitada wrote:
> snip
>
> "It was argued by Schrodinger and Pauli that such a time-energy relation
> should be a straightforward consequence of a commutation rule with the
> structure...
>
> [T^hat, H^hat] = i hbar I^hat (5.8)
>
> which is the commutation rule between the time operator T^hat and the
> Hamiltonian H^hat representing the variables t and E. Does there exist
> such a time-operator T^hat within the usual QT? The answer is no, for
> the following reason... [given the position-momentum relation [x^hat_i,
> p^hat_i] = i hbar I^hat, i = 1,2,3] ... it follows that there should
> exist the relation
>
> i hbar * (\partial f(T^hat))/(\patial T^hat) = [f(T^hat), H^hat] (5.9)
>
> which is completely equivalent to (5.6). [i hbar * (\partial
> f(x^hat_i))/(\patial x^hat_i) = [f(x^hat_i), p^hat] , i = 1, 2, 3 (5.6)]
> The application of (5.9) to the unitary operator
>
> f(T^hat) = exp{i \alpha T^hat} (5.10)
>
> leads to
>
> -hbar \alpha f(T^hat) Phi_E = Ef(T^hat)Phi_E - H^hat f(T^hat) Phi_E,
> (5.11)
>
> where \alpha is a real number and Phi_E an eigenfunction of H^hat with
> eigenvalues E:
>
> H^hat Phi_E = E Phi_E . (5.12)
>
> With
>
> Psi_E = f(T^hat)Phi_E, (5.13)
>
> we have
>
> H(hat) Psi_E = (E + hbar\alpha)Psi_E
>
> 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."
[MP]
The continuity of the energy spectrum follows from the requirement
that states are *localizable in time coordinate*. One can however
give this assumption and as Hitoshi showed one can time operator
in the subspace of energy eigenestates. The multiplication with t
however does *not* leave energy eigenstate to a superposition of
energy eigestates but takes out of this subspace. Time operator
takes you outside the system!
Discrete energy spectrum follows from the assumption that some field
equation is satisfied. For instance, for free Schrodinger equation E=
p^2/2m m where p is momentum of planewave ezp(iEt-ipx) (hbar=c=1). For
given momentum p only single energy is allowed.
The physical content of discrete energy spectra is probability
conservation. If there where no constraints, Psi could exist
in finite time interval: system would be created and destroyed.
And also determinism. Once you have linear field equation you have
determinism and constraint on energy spectrum.
[By the way, cognitive spacetime sheets, which exist finite period
of time are indeed reflections of nondeterminism of *nonlinear* field
equations associated with Kaehler action.]
Best,
MP
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