On Fri, Feb 06, 2009 at 08:59:44AM -0500, Jesse Mazer wrote:
>
> Ah, never mind, rereading your post I think I see where I misunderstood you--you weren't saying "nothing in QM says anything about" the amplitude of an eigenvector that you square to get the probability of measuring that eigenvector's eigenvalue, you were saying "nothing in QM says anything about" how the length of the state vector immediately after the measurement "collapses" the system's quantum state is related to the length of the eigenvector it collapses onto (since the probabilities given by squaring the amplitudes of the eignevectors always get normalized I think it doesn't matter, the 'direction' of the state vector is all that's important).
>
> Still, I don't quite see where Mallah makes the mistake about the Born rule you accuse him of making, what specific quote are you referring to?
>
> Jesse
>
According to Wikipedia, Born's rule is that the probability of an
observed result \lambda_i is given by <\psi|P_i|\psi>, where P_i is the
projection onto the eigenspace corresponding to \lambda_i of the
observable.
This formula is only correct if \psi is normalised. More correctly,
the above formula should be divided by <\psi|\psi>.
This probability can be interpreted as a conditional probability - the
probability of observing outcome \lambda_i for some observation A,
_given_ a pre-measurment state \psi.
What is important here is that it says nothing about what the state
vector is after the measurement occurs. There is a (von Neumann)
projection postulate, which says that after measurement, the system
will be found in the state P_i|\psi>, but as I said before, this is
independent of the Born rule, and also it does not state what the
"amplitude" (ie magnitude) of the state is. The v-N PP is also distinctly not
a feature of the MWI (it is basically the Copenhagen collapse).
I think the quote I was responding to was the following:
"In an ordinary quantum mechanical situation (without deaths), and
assuming the Born Rule holds, the effective probability is proportional
to the total squared amplitude of a branch."
If you compare it with the description of the Born rule above (which
computes a conditional probability), there is no sense in which one
can say that "the effective probability is proportional to the total
squared amplitude of a branch" follows directly from the Born
rule. Jacques is assuming something else entirely - perhaps
einselection?
It may be true that if the Born rule is false, then the effective
probability is not proportional to the norm squared (yes I was having
a little dig there, amplitude is a somewhat ambiguous term in this
context, but one could interpret it as meaning norm (or L2-norm, to be
even more precise), but without seeing Jacques's starting assumptions, and the
logic he uses to derive his statement, it is really hard to know if
that is the case.
Cheers
--
----------------------------------------------------------------------------
A/Prof Russell Standish Phone 0425 253119 (mobile)
Mathematics
UNSW SYDNEY 2052 hpcoder.domain.name.hidden
Australia http://www.hpcoders.com.au
----------------------------------------------------------------------------
--~--~---------~--~----~------------~-------~--~----~
You received this message because you are subscribed to the Google Groups "Everything List" group.
To post to this group, send email to everything-list.domain.name.hidden
To unsubscribe from this group, send email to everything-list+unsubscribe.domain.name.hidden
For more options, visit this group at http://groups.google.com/group/everything-list?hl=en
-~----------~----~----~----~------~----~------~--~---
Received on Sun Feb 08 2009 - 02:22:00 PST