# Re: [KevinTryon@hotmail.com: Jacques Mallah]

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

--

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A/Prof Russell Standish                  Phone 0425 253119 (mobile)
Mathematics
UNSW SYDNEY 2052                         hpco...@hpcoders.com.au
Australia                                http://www.hpcoders.com.au
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