On 26/06/2017 3:57 pm, Russell Standish wrote:
On Mon, Jun 26, 2017 at 11:50:45AM +1000, Bruce Kellett wrote:
That is not what is normally meant by the '+' symbol. You have
simply defined a conjunction to be a disjunction!
We are constructively defining +. I would not be so cruel as to use +
if the end point were not the usual group operation.
Yes, the endpoint is that the '+' is simple addition. It seems to me
that if you actually wrote
psi_a v psi_b,
where 'v' stands for disjunction, or 'or', you would not have got very
far with your derivation. By writing the sum
psi_a + psi_b = psi_{ab}
you have, in fact, simply assumed linearity. A significant property of
linear systems is that if you have two solutions, the sum is also a
solution. If you are dealing with sets, the the operation is the union
of sets, which is different. But you specifically state that your
projection operator acting on the ensemble produces a single outcome
psi_a = \P_{a}*psi, so you are dealing with addition of numbers or
functions, not the union of sets.
Thus, for the sum to make sense you must assume linearity. Now linearity
is at the bottom of most distinctive quantum behaviour such as
superposition, interference, and entanglement. It is not surprising,
therefore, that if you assume linearity at the start, you can get QM
with minimal further effort.
Bruce
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