>From: Russell Standish <R.Standish.domain.name.hidden>
> > >From: Russell Standish <R.Standish.domain.name.hidden>
> > >Basically, the i in the equation is to ensure that the Hamiltonian is
>hermitian, which is required by the law of conservation of probability
>(d/dt (psi* psi))=0. This latter law is simply the statement that the axiom
>saying the probability of the certain event is 1, and shall remain so for
>all time.
> >
> > As you know Russell, I find your "derivation" of the SE quite
>wanting. As far as conservation of probability, it is not obvious that
>measure should be conserved as a function of time. In fact, measure is not
>strictly conserved.
> > The i is there to make the equation simpler to write. Of course one
>could write it in terms of real quantities only, such as amplitude and
>phase.
>
>Well wait for the mark 2 version of the Occam paper, where I shall be
>far more explicit in the steps. Sorry it is taking so long, but I have
>a real job to do as well.
OK, I'll try to be patient ...
>One of the Kolmogorov probability axioms is that the certain event has
>probability 1. Conservation of probability is simply asserting that
>the probability of the certain event does not change over time. A
>fairly obvious corrollory one would have thought.
Maybe you're trying to insult my intelligence, but in doing so you only
insult your own.
As I said, measure is not strictly conserved as a function of time. Of
course it is by definition true that the integral of the effective
probability over all observer-moments in all universes (including, since we
are presumably using the AUH, universes that are not governed by QM) and all
times is equal to 1.
Even if it were established that the measure per unit time of an
"observer" was equal to the squared amplitude of that "observer"'s
wavefunction (and this has yet to be satifactorily derived from theory),
that might (in general) not have been conserved. It may well be, too, that
the formula for measure would have to be modified if the sum of square
amplitudes was not conserved - which might even allow conservation of
measure-per-unit-time without requiring conservation of squared amplitudes.
>Saying that the i is there makes it easier to write misses the
>point. The Schroedinger operator is i times a Hermitian operator. This
>is not the most general form of linear operator, so this structure
>ought to have some form of explanation. The one I gave is not actually
>due to me, but I can't think where I first came across it - possibly
>Emile Durand.
It is a very well known fact that a Hermitian Hamiltonian leads to a
unitary (squared-amplitude-sum-conserving) time evolution operator; I don't
know who first said it but it must have been in the early days. Of course,
if it weren't Hermitian, it would have other consequences, such as allowing
consevation of energy to be violated.
BTW, there are some people who attach an almost mystical significance to
the i. I had one teacher who told the class that in classical mechanics, i
is used only for convenience; but in QM it is fundamental; therefore QM is
weird. That's nonsense.
>From: Higgo James <james.higgo.domain.name.hidden>
>Trying to derive SE from AUH is like trying to derive 'Jacques Mallah' from
>AUH.
>It's very easy: all universes exists, so some thoughts of the cleass "the
>SE is -i hbar d/dt psi = H psi" exist.
>Some thoughts of the class "why is the SE -i hbar d/dt psi = H psi" also
>exist - and by WAP we shouldn't wonder why we *are* (not think, but are)
>such a thought. If we weren't, we wouldn't wonder it...
James, you are abusing the AP. SE is not like JM. It would be
worthwhile to try to derive SE because the effective probability of
observer-moments that see it is likely to be large; we might be able to get
a prediction. In the case of JM, one name is just about as likely as
another (of similar length), so most observer-moments probably don't think
they have the name JM.
- - - - - - -
Jacques Mallah (jackmallah.domain.name.hidden)
Physicist / Many Worlder / Devil's Advocate
"I know what no one else knows" - 'Runaway Train', Soul Asylum
My URL:
http://hammer.prohosting.com/~mathmind/
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Received on Thu Jun 29 2000 - 11:38:09 PDT