This video is about the de Broglie hypothesis
which supposedly says matter is a wave as
well as a particle and this equation that
tells us how to find the ‘wavelength’
of a particle.
The issue is, there is a very precise way
to interpret the hypothesis that leads to
interesting questions about quantum mechanics,
but instead it’s often stated in an almost
meaninglessly imprecise form.
Let’s fix that.
This video is a bit more involved then the
previous ones but hopefully not too bad!
But if I fail, don’t worry, you don’t
need to understand all of this video to get
the next.
If you do want to stick around.
I recommend a little revision since it’s
been a while–in particular these videos.
Ok, we’ve got this particle, it’s got
a wavefunction.
Remember the wavefunction of the particle
contains everything we can know about that
particle.
For example, it might say that the particle
is in a superposition of being here, here
and here, and so if you measure it’s position
then it will turn up in one of those places.
There are these special wavefunctions called
position eigenstates.
This means that the particle is really only
in one spot- if you measure it, it will definitely
be there.
Wavefunctions doesn’t just tell us about
the position of a particle, it also tells
us about the speed, or momentum, of the particle.
There is a way to convert a wavefunction from
the position basis to the momentum one.
When you do it you could find your particle
is in a superposition of several speeds.
Here’s the question we’ll focus on: if
the particle is going at one speed, so it’s
wavefunction is a momentum eigenstate, then
where is it?
How do we write that wavefunction is the position
basis?
Is it in a superposition of many places, or
is it in one spot?
If this was every day life, at any point in
time a moving particle is still only in one
spot.
But what we’ll see is, even at one particular
time, a momentum eigenstate is a superposition
of many positions- in fact, all of them, equally.
I’ll show how we motivate this.
As with Noether’s theorem, we do thought
experiments about how a state could be different.
We say, suppose we have some superposition
of locations.
We just imagine, what if these places had
been a bit to the right instead?
Let’s call this offset amount delta.
As you can see, position wavefunctions are
the same but delta to the right.
Remember the way we defined this shift, the
position is changed, but the speeds the particle
is going at is not.
Ok so imagine our original wavefunction was
a momentum eigenstate, what happens to it
when it’s shifted?
We’d expect this wavefunction is exactly
the same right?
Because the shift doesn’t affect the speed,
so it should still be the same momentum wavefunction.
Well actually, that doesn’t mean the wavefunction
has to be exactly the same.
We definitely want it to be going at the same
speed if we measure it.
But there’s still some freedom here.
Remember the probability of a particular speed
is the length of the coefficient in front
here.
Imagine if the state had a negative one in
front of it.
Negative one still has length one.
That’s means the probability of getting
this speed is still 1.
What about if there was an i here That still
has length 1.
In fact, if we go back to thinking of complex
numbers as arrows, all complex number on this
circle has length one.
In physics we call these numbers ‘phases’.
There’s really convenient way to represent
a phases.
Suppose this phase has angle theta.
Using our trigonometry, we could write that
as this complex number.
So we’re just going to define this expression
to be a complex exponential.
Multiplying an complex number by a phase is
the same as just rotating it by that angle.
That’ll be useful.
Getting back to the point,
we require that, when you shift a momentum
eigenstate in position, it doesn’t change
momentum, but it may pick up a phase.
This phase obviously needs to depend on the
amount of shifting delta and it also depends
on the momentum, so those two things should
go into the angle, but then we need a number
that scales how much that the angle changes
as you change delta and p.
We’re going to call it oh I don’t know,
how about... h bar.
Yes the same constant in all those famous
quantum mechanics equations.
This is it’s grand entry- it’s just a
constant there that decides how much a shift
changes the phase.
We can measure it some other ways, and it
turns out it is very very small.
We’re almost there.
But first we’ve got to talk a little bit
more about position wavefunctions.
I’ve talked about about the wavefunction
when it’s a superposition of a few different
spots.
But the thing is, a particle could be in superposition
of a whole infinite range of positions.
What if the particle is in a superposition
of all these places but it’s more likely
to turn up in this area?
How do we represent that?
Before we put a complex number in front of
each position.
Now we’re basically going to do the same,
we’ll assign a complex number to each point
the particle could be, in other words along
the whole line.
Assigning a complex number to every point
on the line- that’s just a complex function
-we’ll call it f(x).
The bigger the complex number, the more likely
the particle will turn up around there.
There are two ways we can draw this complex
function, either, we just draw the real and
imaginary part separately, or draw it 3D,
with the real bit coming out toward you, and
the imaginary bit going up, so all complex
numbers are just arrows.
Both are useful.
For people who’ve seen calculus before,
all I’m saying is that the wavefunction
can be written as an integral of all positions,
where each position is weighted by this complex
function.
If that made no sense to you, don’t worry,
it’s not important.
Anyway, we have this momentum eigenstate,
and we want to write it in the position basis.
We know the position wavefunction is going
to be represented one of these complex functions,
we just need to find it.
Remember that when you translate the momentum
eigenstate, it stays the same, except it picks
up an overall phase.
Then since this function is the momentum eigenstate,
it must do the same.
Well there is a function that fits that bill:
it is this complex squiggly function: where
the blue is the real bit and the red is the
imaginary.
In other words this function is a complex
exponential with this wavelength.
Why is this the function we’re looking for?
Well you can already see it almost has translation
symmetry, the function is the same if you
shift it over a bit... only it’s a bit rotated.
But that’s exactly what would have happened
if we multiplied by a phase!
Remember, multiplying any complex number by
a phase just rotates it by that angle.
So translating is this function is the same
as just multiplying it by a phase.
We’ve now actually derived the de Broglie
hypothesis!
See, all it actually says is, the wavefunction
of a particle with precise momentum is the
complex exponential with this wavelength.
Why the confusion then about the wave particle
duality, where people say a particle acts
like a wave sometimes and like a particle
sometimes.
I think the problem is that for a confusing
time before quantum mechanics, people where
doing things like the double slit experiment
and for understandable reasons, interpreted
them using the wave/ particle duality.
Let me show you an alternative explanation
using all of our current quantum mechnaics.
The way the double or single slit experiment
is done is by firing particles, let’s say
atoms.
Those atoms are set up to have one precise
speed.
In other word’s each atom’s wavefunction
is a momentum eigenstate.
We’ll just look at the real bit, and so
that looks a wave.
If we have a single slit, that wave, after
it passes through the slit, spreads out- just
like a water wave would.
Then we measure where the particles end up.
As you can see, the wavefunction is biggest
in the middle, and so that’s where each
particle is most likely to land.
But that’s what we would have expected even
if we were throwing tennis balls you know?
So people thought, this is purely particle
like behaviour.
The double slit experiment then should be
similar- each particle goes through one slit
and lands most likely right behind that slit.
Instead they got this.
This is how you explain it with quantum mechanics.
Since we don’t measure which door the particle
goes through, each particle’s wavefunction
is a superposition of going through both doors.
This means we add the separate wavefunctions
together.
But interesting things happen when you add
waves.
If the waves were in sync, like they are at
these points because the crests meet other
crests and the toughs meet other troughs here,
the result is a really big wave.
But here, the waves are totally out of sync,
and so there’s no wave left there at all-
they cancel.
The result is that there are these alternating
spots were the wavefunction is big then small
then big again etc etc, to get the right result.
People thought now the particle is acting
as a wave, but wasn’t before.
But as you can see the current interpretation
is much more subtle, but covers both cases
in the same framework.
People also imagined that the particle itself
smeared out into a physical wave, but collected
itself back up as a point particle when measured
at the wall.
Now we say that the particle itself is doing
who knows what while it’s not measured,
if it exists at all, but it’s wavefunction
is the one that spreads out.
Honestly, I’m not sure why people who know
quantum mechanics still talk about the wave/
particle duality at all- maybe because that
seems easier to explain than the wavefunction?
But it annoys me because it makes the world
sound so
paradoxical so we couldn’t possibly hope
to comprehend it.
In the end, maybe that’s the case, but we
won’t know until we try to understand quantum
mechanics.
Anyway, thanks for watching this video- I
know I’ve been away for far far too long.
As a reward for your loyalty, here’s some
homework.
First, What did you think of my supposed derivation
of the momentum wavefunction?
What assumptions did I make that you think
need more justification (there are a few keys
one I saw)
Second, Nils Bohr was a huge proponent of
the wave/particle duality even long after
the modern form of quantum mechanics.
Try and understand his argument for it and
report back.
Ok, see you very soon for the Heisenberg uncertainty
principle.
