Re: Download a new book on quantum mechanics and relativity.
From: Eugene Stefanovich (eugenev_at_synopsys.com)
Date: 09/23/04
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Date: Thu, 23 Sep 2004 16:04:51 -0700
Bilge wrote:
> Eugene Stefanovich:
> >
> >
> >Bilge wrote:
> >> Eugene Stefanovich:
> >>
> >> > I think you should read again about the difference between measurement
> >> >(single act of observation, see subsection 3.2.1) and experiment
> >>
> >> I have thought about it. Your idea of observables and quantum mechanics
> >> as a theory about ensembles is just plain wrong and contradicts every
> >> quantum text I've ever seen. You don't connect theory to experiment
> >> very well at all.
> >
> >Wave function is probability amplitude, right?
>
> Right.
>
> >How do you think the probability is measured if not in ensemble?
>
> One doesn't measure a probability. One measures the outcome of a lot of
> trials, obtains a statistical result and then _infers_ a probability for
> that result in each _trial_. Finally one compares that inferred result to
> the probability obtained from a _calculation_ to see if the two agree. If
> they don't one doesn't adjust the wavefunction to match it. One declares
> the theory bad.
I don't disagree with you here. I think we are talking about the same
things.
>
> >By definition, the probability is the ratio of the number of
> >desirable outcomes of measurements to the total number of
> >measurements.
>
> OK, then let me apply that to a test of quantum mechanics.
> I prepare electrons with their spins along the x-direction.
> I measure the z projection. Using your notion of quantum
> mechanics, I can't know ahead of time what the probability
> of finding the spin along +/-z,
Why not? If you know for sure that electrons' spins were aligned
along the x-axis, then you can calculate that the probabilities of
finding their spins along +/-z are 0.5/0.5. If you do not know
how exactly electrons were prepared, you should measure.
> so I say that the probability
> of +z is P_up and -z is P_down. I now count 25 particles with
> +z and 75 paricles with -z.
That's fine so far, except that your ensemble is very small
(just 100 copies) and the measured probability will have a substantial
error.
> According to you, I've just
> measured the wavefunction, so the wavefunction must be:
>
> |x> = 0.5 |+z> +/- 0.866 |-z>
Here we go: your ensemble was too small and measured frequencies
of +z and -z didn't come out equal as they should. You need to
increase your ensemble, preferably make it infinite, in order
to measure probabilities accurately and compare them with theoretical
prediction.
>
> Therefore quantum mechanics tells me the right answer. Always. Does
> that sound very reasonable to you? I hope not.
I don't think I understand you here. Yes, quantum mechanics always
tells right answer, but I suspect, that's not what you wanted to say.
>
> Do you think perhaps that quantum mechanics might tell you the
> probability amplitude for the wavefunctions _before_ the experiment, so
> that you are comparing the probability given by quantum mechanics to the
> _statistics_ of an experiment?
I am still not getting your point. If I knew before experiment that
electrons' spins were oriented along x, then I could use QM to calculate
the probabilities of measuring z-projections, and to compare this
prediction with actual experiment, and find out that theory and
experiment agree with each other. If I didn't know how the electrons in
the ensemble were prepared, then I can perform experiment, and get some
useful info about the electron's wave function. If I measure frequencies
for +z and -z, I can find out (in the limit of infinite number
of measurements) the squares of the wavefunction components in the
z-representation. This is not full wavefunction, but better than nothing.
>
> Do you know the difference in a _probalistic_ theory and how it
> relates to a _statistical_ experiment?
>
> >So, in order to
> >have probability you need to have many (preferably infinite number
> >of) measurements performed on identically prepared systems. You need
> >to have an ensemble.
>
> Oh gee. You've just pointed out some real problems here. According to you,
> knowing the probability amplitude requires me to measure the probability
> amplitude.
I am missing your point again. If you know something, you are not
required
to measure unless you want to confirm that what you know is correct.
You cannot measure directly the probability amplitude. You can
measure probability.
> Does that mean the probability for the first particle to be
> in a given state depends upon the wavefunctions of each of the particles
> that come after it?
Where did I say that???
Eugene.
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