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