On Wed, Nov 04, 2015 at 03:15:51PM +1100, Bruce Kellett wrote:
> On 4/11/2015 1:26 pm, Russell Standish wrote:
> >On Wed, Nov 04, 2015 at 11:59:41AM +1100, Bruce Kellett wrote:
> >>I disagree. I do not think the quantum mechanics /ab initio/ is in
> >>any way possible.
> >This is what I do in appendix D of my book. It then behooves you to
> >point out where exactly that is wrong.
>
> You don't actually derive QM /ab initio/ there: you know where you
> want to get and then make a series of appropriate assumptions. You
It is not illegal to have a target in mind when deriving
something. The important point is not to sneak the target into the
assumptions. Provided the assumptions stand on their own, that's fine.
> postulate that the observer can choose an observable, which has a
> discrete set of possible outcomes.
Yes - I can't image what observation means otherwise. Even when
measuring a supposedly continuous variable, the outcome of the
measurement will still be discrete - such as somewhere in the discrete
interval 1.345 +/- 0.001.
> You then assume a probability
> interpretation.
The axioms are there to give a definition of the term "probability". I
haven't heard of them being controversial.
> The relationship to observations is simply assumed.
Given that observation is a matter of turning possible into actual,
or selecting a particular measurment outcome from the range of
possible, there will be a probability associated with each possible outcome.
> I do not see any derivation of the fact that possible outcomes of
> measurements are the eigenvalues of the corresponding operator.
That the outcomes of observables are encoded as eigenvalues is a
convention that has some calculation convenience, but no further
significance. One can formulate QM entirely in terms of projection
operators without the need of the notion of Observable being a
hermitian operator with eigenvalues being the set of outcomes.
To get from one to other, in my derivation you have a set of
projection operators {Pₐ}, where ∪ a = S, the certain event. The
observable A is simply given by summing over the projection operators:
A = ∑ a Pₐ
The reverse decomposition of A is easy also - and goes by the name of
Gram-Schmidt orthonormalisation.
> You
> have a major basis problem because the best you have is that a state
> (vector) is a linear sum over some complete basis set, but you don't
> know what that basis set might be, or what it might represent.
>
That comes from the observer's choice. Not a problem if the observer
actually has free will. A bit more of a problem if you want the
observer's choice to be determined by the physics.
> The derivation of a time evolution equation depends on the
> assumption of a classical time variable, which you assume is uniform
> and universal.
It neither assumed uniform nor universal. It is a subjective notion
(making sense only to the observer), satisfying the axioms of a
timescale.
However, to get the usual Schrӧdinger equation, your timescale needs
to be the real numbers. Quite whether that difference makes a
difference physically, I leave to the interested student :).
> You do not demonstrate that the 'Hamiltonian' you
> have in your time evolution equation is the energy operator.
It is by virtue of Noether's theorem. In my work I don't address the
correspondence principle at all, but if it's going to be addressed,
then Vic Stenger's gauge invariance approach is most likely to be how
it is done.
> In fact
> you show nothing at all about H, or about any dynamics -- which
> presumably you take over from classical mechanics or something
> similar.
>
Not at all. I stop before anything to with classical mechanics, or the
correpondence principle comes into play.
> This scarcely counts as a derivation of any useful physics at all,
> much less of quantum mechanics that relates to observational
> results.
>
It is a derivation of quantum mechanics, from simple assumptions about
what it means to observe stuff. That quantum mechanics needs
a correspondence principle to make contact with classical measuring
apparatuses goes beyond what I was attempting to do.
--
----------------------------------------------------------------------------
Prof Russell Standish Phone 0425 253119 (mobile)
Principal, High Performance Coders
Visiting Professor of Mathematics [email protected]
University of New South Wales http://www.hpcoders.com.au
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