To
better understand the maths, let’s get familiar with at least one
experiment first to get a picture in the mind.
Young's double-slit experiment with electrons
The set up is just to put a traditional
double slit in the path of an electron beam, shot out from an electron gun to
see if there would be interference in the results or not.
Picture from Wikipedia
Historically, the issue of waves vs
particle nature of things started all the way back to Newton. Newton thought
that lights are particles, perhaps due to geometrical optics where you can
trace the path of light through lenses by just drawing straight lines. There is
also a common-sense answer (which ignores how small light's wavelengths are)
that if light is a wave, how can our shadows be so sharp instead of blurry?
Thomas Young back in 1800s first did the
double-slit experiment on light. It's basically the same set up as the picture
above, just replace the electron gun with a light from a lamp, which is focused
via a small hole. Laser hasn't been invented yet then. As light passes through
the double slit, if it is made out of particles, we should only see two slits
of light at the screen, yet we see an interference pattern!
Wait a minute you might say. You go get
a torchlight, cut out two slits out of a cardboard and shine the torchlight
through the slits, you see two slits of light shining through. Where is the
interference pattern? The caveat for the double-slit is that the size of the
slit and the distance between the slit should be roughly around the wavelength
of whatever waves you wish to pass through it. And the wavelength of light is
around 400 to 700 nanometres. For comparison, the size of a bacteria is about
1000 nanometres. The enlarged slits in the picture are merely for illustration
purposes, it's not to scale.
What can produce an interference
pattern? Waves. Observe the gif below. Waves can meet with each other and if
they happen to be in phase at the position where they meet the screen,
constructive interference happens, the amplitudes add up and you see light-gathering
there. If they happen to have opposite amplitude at another position,
destructive interference happens and you are left with a dark region.
Destructive interference is also what happens when you use noise-cancelling
headphones.
gif from wikipedia
So Thomas Young settled that lights are
waves after all, with wavelengths being very small, thus our shadows seem
sharp. Next up, Maxwell showed that light is electromagnetic waves with a
calculable theoretical speed. Thus it was with great difficulty to accept again
that light maybe particles in some other situations. That's why Planck didn't
believe the mathematical trick he did had a physical significance. And Einstein
was pretty much didn't get much support when he took the idea of photon (light
as particles) seriously.
Louis de Broglie had some idea that if
waves have particle-like properties, might not particles also behave like
waves? It took a long time, but finally, the proper experiment was done using
electron beams fired from electron guns towards the double slit only to find
(to no one's surprise by then) that yes, electrons exhibit interference pattern
too.
What's so hard for classical thinking
and expectations to accept is that a thing is either a particle or a wave. How
can it exhibit particle-like behaviour in some cases and wave behaviour in
other cases just for the convenience of explaining what happens in certain
cases? Quantum thinking would have to accept a certain relaxation of this
criterion that a thing must be either a particle or a wave. So it could be that
they have both properties which are real (as advocated by Bohm's
interpretation), or that they behave like wave or particles depending on how we
set up the experiment (Copenhagen interpretation). Or some other possibilities.
It's a common practice to not be too concerned with our language to say it's a
particle-wave. Usually, we just use the term particle and the wave properties
are understood to be there when needed.
Let's take a breath here to reflect that
you might not find the results so far as strange at all. I had to point out
what kind of thinking (classical) would make these results weird. If you had at
all heard that quantum physics upends a lot of classical notions, you would
have already come in, prepared to have an open mind and not be attached to
classical thinking. So you readily see nothing weird about quantum physics,
just a different set of rules. You might be gradually be used to the quantum
logic pathway to make sense of quantum, which are called the modal interpretations.
Continuing on the double-slit
experiments, there are quite a few additions to the basic experiment to exhibit
some other properties of quantum systems.
First, the experiment can be done with
single particles. A single photon, or single electrons or other particles.
Single as in the particles gets shoot through the slit one by one. If it passes
the slit, we use a super-sensitive detector, capable of detecting one particle
at a time and also recording the position of where is the particle detected.
Over time, the interference pattern can be seen to be build up again. One by
one, the particles somehow knows where to land in order to rebuild that
interference pattern.
It gives a creepy feeling for people to
think that somehow a single particle has to use its wave properties to feel
both slits in order to land at the positions which is consistent with the
interference pattern. So a particle can interfere with itself! Different
interpretations will give different pictures of this phenomenon. So don't be
attached to the first two sentences of this paragraph!
Second variation, we can try to observe
which path did the particle took on its way to the screen. There are many
subtle details and recent developments in this bit, elaborated more later on
when we discuss wave-particle duality.
For now, the simplified version is if we
put a measurement device to detect if the particles would go through one slit
or another. As long as we can have the information of which path, left slit or
right slit was taken by the individual particles as they pass, we see no
interference pattern; the particles make a pattern of two slits on the
detector.
For most Buddhists, this is likely not
the first time you had heard of this double-slit experiment and you might be
very eager to see the one thing you are interested from the popular telling of
this experiment. The act of observing things (with or without consciousness
involved is interpretation dependent) changes what happens to the thing you
observe. Do take note that the observation need not necessarily involve
consciousness and the most important thing is the measuring device is present.
Also, we shall see this property that measurement changes quantum systems even
in the Stern-Gerlach experiment later. The big technical name you can pin to
this behaviour can be called contextuality. More technical treatment of
contextuality follows later.
Perhaps the most important take away is
that do not place all your eggs onto one interpretation yet, just because of
preconceived notion that it fits in with Buddhism (we shall see if it does and
how it does). Have some patience and an open mind to keep on reading and
participate in the analysis. As per the spirit of Kalama sutta, there are three
main ways of deciding what to believe, revelation, reasoning and experience.
The experience part is this section of experiment. The reasoning shall be done
in the analysis, revelation is basically all the physics other people had
discovered which you are soaking up now. As the experience part is most
important in Buddhism, do place the same importance of it in physics. An
interpretation of quantum mechanics means it currently has no way to
experimentally distinguish itself from other interpretations, or the
experiments done to do so had not been thought of yet, or it is not yet
technically feasible, or it was done but not universally conclusive and
persuasive yet. So no point to attach to one viewpoint (interpretation) based
on the notion: it agrees with my view.
We shall move on to the mathematical structure of quantum and
includes introducing the axiom of quantum as taught to physics undergraduates
even now. Many of the terms are repeated there, so don’t
worry. You’ll get a better picture of the maths there.
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