What we’re covering today is a rule of quantum
mechanics we haven’t met before that is
responsible for a lot of the weirdness.
It explains the Heisenberg uncertainty principle-
but even more importantly, explains why a
superposition is different from not knowing
what an object is doing.
I know I’ve been away for a while, so if
you need to refresh on any of the old stuff,
do that first, we’ll need it.
If anything is confusing in this video, let
me know, because I’ll do a few follow up
videos trying to make this idea clearer -it’s
kinda an important one.
Say there’s some object you’re interested
in.
Then there are all these things you can ask
about what it’s doing.
For example, where is it, how fast is it going,
how much is it spinning?
etc.
We’ll call these things you can measure
about the object its observables.
Usually we think an object has a value for
each of these, but quantum mechanics instead
says, each object is in a superposition of
all the values it can have.
In the regular rules, different observables
can be independent, for example position and
speed- this means that knowing an object’s
position doesn’t say anything about how
fast it’s going.
So if you want me to tell you what the object
is doing, it’s clearly not enough for me
to just say where it is.
Now, how can we translate that to quantum
mechanics?
Say I have a particle and you’re interested
in observable X for that particle, again,
that can be position.
I can’t tell you where the object is- the
best I can do is tell you the wavefunction,
right?
Because if you know that, you’ll know the
probabilities for all the places it can turn
up when measured.
But say you’re also interested in some other
observable Y, again, this could be the object’s
speed.
You know that that must be in a superposition
as well, so you want to know what that wavefunction
is.
But X and Y are independent, and so there’s
no way 
to know without asking me.
You might be thinking that this difference
between classical physics and quantum mechanics
is a slight technical curiosity that only
a physicist could get excited about.
Fair enough.
But I hope to show you that it has some pretty
amazing consequences.
Firstly, it helps us understand superposition.
We said that, while nobody is looking, an
object is in a superposition of all its possible
states for observable X.
When we measure X, it will collapse to one
of those states, with probability given by
the square of the coefficient- according to
quantum mechanics.
But this seems so complicated.
Why shouldn’t we believe it’s actually
just in one state or another, with the right
probabilities- but we just didn’t know which?
Both these explanations, the quantum mechanics
superposition craziness one, and the regular
sensible one both predict the same thing-
so shouldn’t we believe the simpler one?
The answer is, sure, you can’t tell the
difference between these two when you measure
X.
But if you measure Y instead, these two theories
will make different predictions- and only
quantum mechanics gets it right.
We’ve met this once before, and it’s called
interference, but we’re finally ready to
explain it properly.
To illustrate all this, I’m going to use
a toy model that’s nice and simple, then
I’ll explain how to do it in the really
interesting cases later.
Say there’s an observable called the up
or down-ness of a particle, when you measure
it, every particle is either up or down, so
in general quantum mechanics tells us they
are in a superposition.
But there’s another property of a particle
that you can measure, called its left or rightness,
and similarly the particle can be in a superposition
of left or right.
I told you earlier that I should be able to
rewrite the wavefunction in terms of any variable.
But how do I convert from the wavefunction
in terms of up/ downess to left/rightness,
or in general, how do I go from observable
X to Y?
First, let’s call each of these states the
eigenstates of X.
My task really is to rewrite these in terms
of eigenstates of Y – and there is always
a set rule to do this.
In our case:
Up equals right plus left,
and Down equals right minus left.
I’ll explain what motivates these conversions
when I do some real examples.
But back to this toy one, if I had a particle
that was up, and I decided to measure it’s
left or rightness, take a second to predict
what would happen, and predict the same for
a down particle.
The answer is, for both cases, there is a
50% chance of left or right.
Now here’s the tricky bit.
What if I had a particle that was in a superposition
of half up plus half down?
If I measured it’s left or rightness, what
would happen?
Think about it, and then click one of these
options below
