Hitoshi Kitada (email@example.com)
Mon, 6 Sep 1999 19:06:06 +0900
This is a typographical correction to [time 692]:
exp(-itH) should be read
in the followings.
----- Original Message -----
From: Hitoshi Kitada <firstname.lastname@example.org>
To: Stephen P. King <email@example.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
> For the state vector Psi(t) of an LS, say L, the Scroedinger equation holds
> 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
> 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,
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