Background and
Interpretations of Quantum Mechanics
I
hope in my series on Wave-Particle Duality I have convinced
you that the wave equation for matter exists, or at least that it is not a
complete crazy concept. However, I was
deliberately vague about what this wave equation actually means, besides only
telling you that everything propagates like a wave. That means that particles such as electrons,
photons, etc, have the properties of interference and diffraction, and so can
be described by a wave equation, but what does that wave equation actually
MEAN? We can create a concept like mass
times velocity times acceleration squared divided by the speed of light, but
that concept is rather meaningless to us.
Does the wave equation actually tell us anything, or is it a bunch of
meaningless math? Well, physicists
expected that it meant something- it had to be significant somehow that
particles are described by a wave equation, so they had reason to believe it
wasn’t idle mathematics.
Physicists
like to decide difficult questions like that by performing experiments, and so
they designed an experiment in which they shot single photons through a very
small slit, and they found that even when only one photon at a time went
through, a diffraction pattern was created as they marked where each photon
landed over a period of time, which means that probability of a photon landing
at certain location is related to the wave equation. In places where there’s a “peak” in the wave
equation, they found there was a high probability of photons landing there, and
in places where there was a “trough” in the wave equation, there was a low
probability of photons landing there.
Why are these probabilities?
Well, because for any given SINGLE photon, physicists cannot, from the
wave equation, determine exactly where it is going to land- they can just tell
you that a lot of similar photons landed over here, and very few similar
photons landed over there, but that single photon could still land over there.
Imagine
that you have two friends, who have a rope stretched between them. One of your friends starts shaking the end of
the rope up and down, creating a wave.
Now, this wave isn’t EXACTLY anywhere- it’s spread out over the distance
of the rope it’s traveled down. It
doesn’t have a precise, single-point location.
However, looking at the wave, you can define a wavelength- the distance
between one peak to the next. That isn’t
too hard. Now pretend your friend stops
shaking the rope and it becomes stationary.
Now he gives the rope a single jerk.
This time, you can define a precise location for the wave- for example,
the location of the maximum amplitude in the pulse. However, defining a wavelength is impossible-
there’s only one maximum, and you can’t define a distance between only one
thing! Of course, you can have
trade-offs in the middle, with multiple pulses, where it’s sort of possible to
define a wavelength and sort of possible to define a location, but you cannot
give both precisely. This, of course,
applies to ALL wave phenomena, and is similar of the uncertainty principle that
so many people dread. Eventually I’d like
to post a series that gives this more precisely, but the math is horrendous-
for now you’ll have to accept that this general concept applies to the Schrödinger
wave equation also. The important point at
the moment is that the Heisenberg Uncertainty Principle can be DERIVED under
quantum mechanics- it is NOT an observed phenomena, as some books imply.
Since
the wave equation describes the position of the particle, one would expect one
of the uncertainties to involve position, and if you remember the de Broglie
formula from Electron Waves, you know that the
wavelength of an electron wave is related to the momentum, and hence the
velocity. So, based on the reasoning
above, we have what is by far the most famous formulation of the uncertainty
principle: the more precisely you know the position of the particle, the less
precisely you know its velocity.
However, there are others, and for example, the more precisely you know
a particle’s energy, the less precisely you know the time. The big question, though, that physicists
wanted to answer was: do those “you know”s that I put into the last sentence
belong there? Is it merely that we
cannot KNOW both the position and velocity precisely, or is it that there
DOESN’T EXIST a precisely defined velocity and position? Today, quantum mechanics unequivocally answer
the latter, but most people don’t understand why, and that’s what this series
is going to attempt to explain.
In
the early days of quantum mechanics, physicists weren’t certain of the answers
either. They tended to fall into three
groups:
1) Realists. These people believed that particles have
well-defined positions, velocities, etc, and that we just have limits on what
we can know. Regarding the location of a
particle, it has a well-defined position, but quantum mechanics is not
complete, and that is why the wave equation gives us probabilities.
2) The orthodox position. (It was clearly named later.) There actually really is an uncertainty in
the position/velocity of the particle- the position IS NOT well-determined and
we just cannot know it. Somehow or
other, observing the particle makes it collapse into a specific position
(remember from wave particle duality that every transfers energy like a
particle), but before that time there was NOT a well-defined position. These uncertainties are really “real”, and
not just epistemic.
3) The agnostics. They thought this argument didn’t belong in
physics, but instead philosophy. Since
this is an argument about things that aren’t observed, we cannot know anything
about them…or so they thought. The EPR Paradox, however, changed that
completely.
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