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 hpco...@hpcoders.com.au 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-l...@googlegroups.com To unsubscribe from this group, send email to everything-list+unsubscr...@googlegroups.com For more options, visit this group at http://groups.google.com/group/everything-list?hl=en -~----------~----~----~----~------~----~------~--~---
