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Similarly if you have a light beam that's weak enough going over
by a single photon's worth of energy, one at a time, with a double slit experiment, you would see the exact same thing
you can only observe one photon's worth of light energy at a time
so it comes up as little blips one after another
it's only when you build up a whole range of them, just like for the electron experiment
that you see this overall pattern
so this led to the following interpretation
so call
the Copenhagen interpretation, because that's where the conference was held
the waves then becomes associated with the probability of detection at that point
if you have a big amplitude of the wave there, that means it is more likely
for you to be able to absorb or detect that particle
and so it becomes this weird notion that's basically at the crux of the wave-particle duality that
particles and light, everything
propagates and move like waves, in the sense that
there's probability of detection over a large range of space
but then when it comes time to detect it, they exhibit the particle behavior where you just one blip
at a particular point
and where it actually ends up is completely random and statistical
as far as we know, years and years of trying to search for something more meaningful has not produced anything
appearently nature is this weird statistical beast that we have to deal with
but due to certain mathematical relationship, it turns out that
the wave function, so to speak, doesn't give probability directly
but similar to how lights you can't add up the power of two waves to give the overall power of the waves because of interference
the wave is, first of all, represented by the electric field amplitude
and the electric field amplitude square is what gives you the intensity, the power which is what you observe
which allows for interference before mixing it all up together
similarly ,the probably of
detecting a certain particle at x, y, z, and at that time is directly proportional
to the magnitude of the wave function squared, of course, it also depends on x, y, z and time
now I put these absolute value signs there, because once the math works out
it turns out that this wave function is inherently complex
so we have to do the wave function times its complex conjugate
and Schrodinger was the first person to write down
a wave equation for such these matter waves, which is, of course, called "Schrodinger's equation"
now I am probably completely butchering the pronounciation of "Schrodinger"
but stick it into google translate if you want the proper pronounciation and we will just go on with the math
so at this point, what we know about the matter wave?
well we know that from De Broglie's work, we can write
the energy of a particle is relate to its frequency
and just for symmetry's sake
we often like to write that
where h_bar is ...
and also the momentum is related to the wavelength which allows is to write this k being the wave number
and so you can see where we are driving at, because once again we are going to start from
what we expect for the answer and see what kind of behavior it plays well with
our go-to solution
is once again the plane wave, we are going to stick with 1D for now, so it is in x and t
therefore we have some amplitude e to the i, basically (kx-wt), so I have
...
dividing up the h-bar
this would tend to describe a particle that is traveling freely in space
so when you have a plane wave traveling in a certain direction
in this empty space basically
that hints that this will represent a particle traveling freely without any potentials around it
so let's jiggle this around until we get something that looks like a wave equation
First off, we are going to take the time derivative
...
...
of the same exponential, if we take the spatial derivative
you get ...
ok, so then, we need some way to relate these two relationships
and for a non-relativistic particle
we know that the energy, in this case once again, because we don't have any potential
around the particle as it is just moving freely ...
so we can make this relationship: ...
In order to get a P^2, we need to derive this again
...
...
and this up here, we rewrite as ...
...
now solving for
P^2 * psi, an equating both sides
we end up with
this lovely equation which relates our spatial derivative with our temporal derivative, however
something is amissed here
first of, you have a double spatial derivative
but a single temporal derivative
and
there is complex factor of i out here, where h- bar is real, m is real, but there is an i here
both of these, because of the single derivative and the i
so the solution actually has to be
inherently complex, so no more of this real operator
making it a cosine
no more
and that's why when we are dealing with
working out the probability, we can't just take the square of the function
we have to take itself times its conjugate
and when you do that, just want to quickly point out that, any phase shift in the wave, because it is the amplitude that matters
the phase shift actually don't have very much observable effects
and this brings us to a close for our story as we have
come all the way to the Schrodinger's equation was describes these matter waves for a 1D free particle
there is one little additional step when we have potential energy involved
but this is basically
how we've come about and made these matter waves, and starting with this wave mechanics
and evolving into quantum mechanics as we know it