Re: IS this for real?!

From: Matt Mahoney (matmahoney_at_yahoo.com)
Date: 08/18/04 

Date: Tue, 17 Aug 2004 22:14:14 GMT

"Mok-Kong Shen" <mok-kong.shen@t-online.de> wrote in message
news:cfshkr$dj$03$1@news.t-online.com...
>
>
> Douglas A. Gwyn wrote:
>
> [snip]
> > the Heisenberg uncertainty principle, which is a
> > direct consequence of a standard fact about Fourier
> > transforms.
>
> Could you kindly elaborate this a little bit in a way
> that a layman can understand? Thanks.
>
> M. K. Shen
>

Suppose you have a sine wave (e.g. sound) that varies in freqency over time,
and you ask what the frequency, f, is at some instant in time, t. If you
take a broad time window around t, you can measure the frequency fairly
accurately by counting the number of cycles, but there is an uncertainty
about t (call this dt). If you decrease dt, then you increase the
uncertainty, df, in the frequency. The uncertainty principle says that the
best you can do is df*dt >= 1.

In quantum mechanics, there is another (less intuitive) Heisenberg
uncertainty principle which says that you cannot know the position and
velocity of a particle with accuracy greater than Planck's constant, h,
about 10^-34 Joule-seconds. The same rule applies to any other combination
of measurements with the same units, such as angular momentum. In fact, if
you measure the spin of a proton, neutron, or electron, you will always get
exactly h/2 or -h/2 along whatever axis you decide to use in your
measurement. These correspond to the spin axis aligned with your chosen
orientation and spinning in one direction or the other.

According to quantum mechanics, the angle between the actual spin axis and
your detector determines the probability of the two possible outcomes. If
the spin axis is at 90 degrees to your detector, then the outcome is random.
(I believe the probability is (1 + sin a)/2). You can generate such
particles and confirm this experimentally for any angle and get results in
agreement with quantum mechanics.

Now if a particle with 0 spin (say a neutral pion) decays into two particles
with spin 1/2 (say a positron and electron), then the classical law of
conservation of angular momentum says that the spins must have exactly
opposite orientations. The orientation of one particle is random, and the
second is exactly opposite. You can confirm this experimentally. Now if
you place two detectors with their measurement axes parallel, you will
always observe opposite spins, no matter how you orient the pair of
detectors.

Quantum mechanics doesn't offer a satisfying explanation for this. If both
particles are at, say, 45 deg. to the detector, and the detectors are
independent, then there should be some nonzero probability that both
measurements would show spins in the same direction. But this never
happens. If you know what happens at one detector, then the other outcome
is completely determined, as if the particles are always emitted with spins
parallel to the detectors. It can't be the case that the detectors affect
each other somehow, because that would require transmitting information
faster than the speed of light. A variation of this experiment is known as
Bell's inequality.

More generally, quantum mechanics defines a wave equation whose density
determines the probability of detecting a particle. For example, you can
compute the electromagnetic wave around an antenna, and quantum mechanics
will tell you the average rate of photon observations. However, the theory
only works if you know nothing else. The theory is incomplete because if
you know something about other particles, then you can use classical
mechanics to constrain what you might observe. We say that these particles
are entangled, but if quantum mechanics is truly a probability function,
then there doesn't seem to be a satisfactory explanation for how
entanglement works.

I hope my understanding is correct.

-- Matt Mahoney

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