# [time 693] Correction to [time 692] re [time 686] Time operator?

Hitoshi Kitada (hitoshi@kitada.com)
Mon, 6 Sep 1999 19:06:06 +0900

This is a typographical correction to [time 692]:

exp(-itH) should be read

exp(-itH/h)

in the followings.

Best wishes,
Hitoshi

----- Original Message -----
From: Hitoshi Kitada <hitoshi@kitada.com>
To: Stephen P. King <stephenk1@home.com>
Cc: <time@kitada.com>
Sent: Monday, September 06, 1999 6:58 PM
Subject: [time 692] Re: [time 691] Re: [time 690] Re: [time 689] Re: [time
688] Re: [time 687] Re: [time 686] Time operator?

> 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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