**Hitoshi Kitada** (*hitoshi@kitada.com*)

*Mon, 6 Sep 1999 19:06:06 +0900*

**Messages sorted by:**[ date ] [ thread ] [ subject ] [ author ]**Next message:**WDEshleman@aol.com: "[time 694] Re: [time 674] Reply to NOW/PAST question"**Previous message:**Hitoshi Kitada: "[time 692] Re: [time 691] Re: [time 690] Re: [time 689] Re: [time 688] Re: [time 687] Re: [time 686] Time operator?"**In reply to:**Stephen P. King: "[time 686] Time operator?"**Next in thread:**Stephen P. King: "[time 695] Re: [time 691] ... Re: [time 686] Time operator?"

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
*

*>
*

*>
*

**Next message:**WDEshleman@aol.com: "[time 694] Re: [time 674] Reply to NOW/PAST question"**Previous message:**Hitoshi Kitada: "[time 692] Re: [time 691] Re: [time 690] Re: [time 689] Re: [time 688] Re: [time 687] Re: [time 686] Time operator?"**In reply to:**Stephen P. King: "[time 686] Time operator?"**Next in thread:**Stephen P. King: "[time 695] Re: [time 691] ... Re: [time 686] Time operator?"

*
This archive was generated by hypermail 2.0b3
on Sat Oct 16 1999 - 00:36:39 JST
*