# [time 692] Re: [time 691] Re: [time 690] Re: [time 689] Re: [time 688] Re: [time 687] Re: [time 686] Time operator?

Mon, 6 Sep 1999 18:58:30 +0900

Dear Stephen,

The local time t can be thought as a canonical conjugate to H in the following
sense:

For the state vector Psi(t) of an LS, say L, the Scroedinger equation holds
identically:

h d
- -- Psi(t) + H Psi(t) = 0. (h being the Planck constant/(2 pi) )
i dt

(Recall that the local time t is defined so that this equation becomes the
identity. I.e. the local time t is defined as the exponent t of exp(-itH).
Thus the state vector

Psi(t) = exp(-itH) Psi(0)

of L (with the initial state Psi(0) ) automatically satisfies the above
equation.)

Define an operator T equal to

h d
- --
i dt

Then the Schroedinger equation (identity) becomes

T Psi(t) + H Psi(t) = 0.

Thus every state vector Psi(t) of L is identically a solution of the
Scroedinger equation with Hamiltonian H, and we have on such states

T = -H. (*)

This T clearly satisfies

i[T, t] = h.

In this sense, T is a canonical conjugate to t. H is related with T by the
above relation (*), which holds identically on the state vectors of L. This
means H (restricted to the space of the state vectors of L) is a canonical
conjugate to t.

Best wishes,
Hitoshi

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