Welcome back
I'm glad you could join me for another go at trying to understand some of the math behind quantum mechanics
Last week I did sort of a introduction to linear algebra and the way that
We saw how you can write out quantum states. And today we're going to expand on that a little bit by
Linking it to finding the probability of a quantum state
Now I just want to say thanks to everyone who left comments on my last video like this
I found them really constructive. Some of you pointed out like mistakes in my pronunciation
but also ways that I could have explained things better and I guess ideas that you had that I didn't include as well.
Shoutout to comment of the day
from someone called Fred who I guess left a lot of really constructive things in a comment, and one of the things he said was
When you were talking about spin it was in a way that made it sound like it had solely to do with magnetic moment.
In fact spin is a carrier of angular momentum,
one of the universally conserved
physical quantities of nature. It is however a mysterious form of angular momentum not being caused by any actual physical rotation.
This is why it is often termed intrinsic spin and
seems to be one of those essentially quantum mechanical effects that can't be understood in classical terms.
So thanks for that comment Fred because I think I yeah
I didn't give like the best explanation of
quantum spin and I think maybe that builds a little bit more on what I said last time.
Here's a little problem to get you warmed up. It's what we're going to work towards understanding throughout the course of this episode,
so don't worry if you don't already understand the math behind it now, but it's the math of the famous thought experiment
regarding Schrodinger's cat. In this thought experiment,
we have a cat stuck in a closed box with a poisoned vial and a Stern-Gerlach analyzer
which can measure the state of a quantum particle.
It can measure the particle to be either
spin up or spin down, and this equation here represents the input state of that quantum particle.
The particle is in a superposition of being spin up or spin down, and if the analyzer
measures the particle to be spin up, then the poison will be released and the cat will die
so the cat is also in a superposition of being alive and dead.
The question is: What is the probability of Schrodinger's cat being alive when we open the box,
given that this is the state describing the quantum particle?
Here's three options for you to choose from just to test your own knowledge and there should be a little option up in the top
right corner to have a vote. Last week
I spoke about
the dot product or finding the inner product between two vectors as a way to find how much overlap is between them.
Now I didn't I guess say in exact terms how to actually find this inner product,
but I want to do an example of that using
this color space example that I used to sort of set things up in the last episode.
Now, you know, say we have these three colors here. We have pink purple and orange
They're made up of different amounts of red, green and blue. And in this case,
It's like a decimal number where the maximum value would have been one. Now to find the inner product between pink and purple,
We kind of expect it to be a reasonably high number and and saying that I mean it's reasonably close to one
these vectors have been normalized and so
that means the length of them the length of the vector is one
So if the overlap between them is one, then they're pretty much perfectly overlapping.
All right, so to find the inner product,
what we're going to do is take the first element of the pink and the purple,
that's how much red they have in them, and times those values together.
So it's going to be 0.7 for the pink times 0.5 for the purple.
Alright, then we're going to add that value from the multiplication to
the second element of both vectors multiplied together, so that's how much green they have.
They both have zero green, zero times zero is zero, adding on zero didn't change much now
Let's add on the product of the third element. So that's how much blue they have in them.
It's point seven for the pink and point eight for the purple.
Adding those three products together for the three elements that make up each color gives us a total value of 0.975
Now that's really close to 1 as you would expect because these pink and purple are very similar colors
We would expect a lot of overlap here.
So an exercise that I went through last time was this one here.
We found the inner product or the overlap between the spin state up and
this unknown state vector psi (ψ) which is defined as some amount of spin up some 'a' amount and
Plus some amount 'b' of spin down and doing that gave us the answer A.
I didn't ask you guys to do this
but it would have been very similar to repeat doing the same thing with the overlap between
Spin down and that same state psi if you went through that one
You would find that the answer is B and I asked you guys to do the inner product between the two
psi states and you would find that the answer for that was 'a squared plus b squared'
Now, what are we going to do with all that? Let's try and do something with what we found here.
So what we're going to say is that this unknown state, and remember we're calling it a 'ket' when it's in these kind of brackets,
We want this state ket to be normalized.
That means you would expect as before that if you had the overlap between something and itself
the answer would be one. One would represent complete overlap
So we already sort of, I guess proved before that the inner product of this state and itself
Is 'a squared plus b squared', but we're also going to set that equal to one saying that it is normalized
so we've kind of given ourselves an identity to work with because we also have
a definition I guess for a and for b
That's what we also worked out before and if we put in our  a  and our  b  into that equation too
This is the I guess final identity that we end up with. It's
Mostly just symbols here
But it becomes quite useful. If you look at it,
each squared inner product term, so that's the thing inside the brackets
corresponds to a possible observed state that was like up or down in terms of spin and
The sum of them if you add them together, it has to equal one and we're asked by brilliant here
What mathematical quantity has properties like this?
This is how probabilities behave.
If something has a probability of one that's like a hundred percent of something occurring if you have two
Possible options or events that could happen and they're equally likely
They would each each have a probability of 0.5 or 1/2
Meaning that if you added those two together, you'd end up with 1 means something is certainly going to happen
But there is a 50% chance of it going one way or the other
That's very similar to what we're dealing with here in the quantum states. So this gets pretty cool
Actually, if you look at this equation here, this is what we've worked up to
and what it means is that
each of these squared inner product terms gives the
probability of observing each of those states,
So I believe the probability of
Observing a spin up state is given by this value here
And this is the probability of observing the spin down state
Now because you know like this is equal to a as we proved before and this is equal to B
that means that  a  and  b  are
Probability amplitudes, that the  a  and the  b  give us the probability
Of being spin up or spin down and that's all wrapped up in this equation here
So we want to use this new thought and put it to the test to see if it can help us explain some experimental results.
So we're going to you finish this quiz and move on to the next one.
So we're looking again at the Stern-Gerlach experiment
You guys corrected me that that's Gerlach, rhymes with 'back', which is probably not what I said last time
and that was this experiment that could
Measure, I guess determine, if a quantum state is spin-up or spin-down
It's using some neutrons coming in here that are in an unknown state and the first analyzer here
Is you know separating them into spin-up and spin-down I believe and then they're passing into a second analyzer
So essentially we're being asked to use those expressions
We've derived for the probability of each observation to figure out
the likelihood of the probability of each of these question mark states
So if the unknown states already had been measured spin up in the first instance, what's the probability
it's going to be measured spin up again or spin down
Now,
You might be able to work this out just by thinking about it, but let's do it with the expressions we worked out
So we worked out the way we would figure this out would be the probability of being measured spin up in this second instance
would be the inner product or the overlap of
Spin up and spin up
If they are normalized we'd expect complete overlap there
And so the probability would be 1 squared which is 1 so pretty much
It's a hundred percent that if something's already been measured to be spin up then if you measure it again
It will also be spin up. That is sort of one of the inherent things about quantum mechanics
Is that when you measure it, it sort of collapses into this state where
continued measurements would give you the same result
So if we wanted to work out the second one
It would be the overlap between being spin up and the first instance and spin-down in the second instance
Because there is no overlap between those two vectors
We've seen that before, they're at 90 degrees to each other and they are two elements of our basis
So almost by definition they have no overlap
And so the probability there is zero squared: zero! This is impossible.
In this next experiment we're going to look at, it's another stern-gerlach experiment
The first analyzer is as before it separates the states
into being spin-up or spin-down,
And then it goes into a second one, which is aligned along of different axis
So it's rotated 90 degrees from the first analyzer
Its magnetic field is aligned along the x axis whereas before it was aligned along the z axis
so when these neutrons are fired through
They're being deflected along a different axis. The magnet is in a different place. So what we see is that when we feed in
Something that's already measured to be in the spin up state
It's split equally into this spin left and spin right
50 is the probability of each of these so it's equally likely to be in either of these two streams
So what we're doing is we're thinking about how we can represent this mathematically. I guess how we write the probability of this happening now
The way that we write out these spin left and spin right states
Is as some linear combination of spin up and spin down. The  a  and the  b  are how much up and how much down
to create the
spin right and then  c  and  d  similarly are the
coefficients, how much spin up and spin down is involved in that spin left state
we know that the spin right and spin left can be written as a linear combination of
some amount spin up in some amount spin down because
what we worked through last time was wanting to have I guess a
Well-behaved vector space so we wanted to set up a basis
That could span, that could create
Any possibility so the basis that we chose was that spin up and that spin down state
We chose them because we could make any
possible state as a combination of using them so we can make
one that points to the left and one that points to the right
using these basis vectors because that's what the basis vectors do they can I guess make any direction.
So this is how we do write it mathematically that the probability of being either here or here
Well, it's 50 percentage. So we write the overlap between
this state and this state is 1/2 and similarly the overlap between this state and this state is also 1/2
it's actually a system of four equations because we also have the
similar experiment of being measured spin down and then being measured either spin left or a spin right.
We're being asked to determine the magnitude of  a, which is I guess how much spin up there is,
the probability amplitude of up in this spin right state. All right, so have a little think about it
We do have three options here on the brilliant quiz that  a  is 1/2, square root of 1/2 or 1
Just have a little like think about it yourself
I do happen to remember the answer to this one from my coursework days
Then the magnitude of  a  is the square root of 1/2. Okay. So apparently 81% of people got that one right.
Ok, so let's have a look at the solution.
This is what we were starting with, the definition of what it means to be I guess spin right and
We knew this from that graphical experiment above, that this inner product is equal to 1/2.
Now if we just kind of combine those two things (ah the LaTeX keeps coming up which is kind of annoying)
If we write out, you know, the definition of spin right
Inside of that probability amplitude definition, you can take the  a  and the  b  out of the equations and we end up with a few more inner products
Where remember if they're both pointing up, full overlap, goes to one. If they're pointing in opposite directions, goes to zero
and so you end up with that or cancelling down to  'a'
Which means that 1/2 is equal to  a  squared because don't forget those whole set of brackets were being squared
So yeah, if  a  squared is equal to 1/2
Then  a  itself must be equal to the square root of 1/2
We can replace the  a  (the unknown) with 1 over the square root of 2
Now 1 over the square root of 2 is exactly the same as saying the square root of 1/2
You're just moving the square root down onto the bottom
because there's no point in taking the square root of the top because that's 1 and the square root of 1 is just 1
So don't let that slight change in notation trick you
Yeah, so really we've written out
This spin left and spin right with sort of one less unknown for the spin left state had
two similar unknowns  c  and  d  which are like  a  and  b
But I guess if you sort of think through it the  c  and the  a  are kind of the same,
so that is also the square root of 1/2, 
so that's been replaced as well.
Okay. So this course thing goes in a little bit more detail, a bit more rigor through finding the value of the other two unknowns  b  and  d
I don't think I'm gonna go through that whole derivation, but the result seems kind of obvious to me
They find that the magnitude of B and D is also the square root of 1/2
B and D do turn out to be going in different directions
So they have a different sign and I think that sort of just comes out of some of this
mathematics. Pretty much you find out that
The spin left and spin rights are made up of equal amount of spin up and spin down which again kind of seems obvious
and I think a similar thing works out if we skip through like
spin up and spin down could be also represented in the same way as being
Equal amounts or equal probabilities of spin left and spin right and that's written in these two equations.
There is one apparently important caveat listed here and they say
you might be tempted to consider the spin-up state as
Describing a set containing 50% of the neutrons in the spin left state and 50% in the spin right state
But this is not the case. The state
Describes a set of identical neutrons each of which is in a state that combines the left and right states
So they're not saying that 50% of the neutrons that go through are in one state or the other, they're saying I guess that every
Neutron is in both of the states
They're in a superposition of both of the states. And that's a bit of a quantum mechanics word
Which means I guess yeah, you are a combination of more than one thing at the same time
So let's put this all together a little bit and do one of the questions. They say consider a beam of neutrons
which has been prepared
in an unnormalized input state
Which can be described as this here three spin up plus four spin down. So
normalize the state vector and
predict the probability of observing it in either the spin up or spin down states.
All right. So let's work through this on paper. I think so for a normalized state if
This is our  a  and this is our  b
Then we would have that  a  squared plus  b  squared is equal to 1.
This is not what we currently have because we have 3 squared plus 4 squared
Is equal to 25?
so
what we want to do is divide each of a and B by a
Normalization constant such as when we do this we want this to equal 1
so if we have
3 over 5
squared plus
4 over 5 squared
That would be equal to 1 so our normalization constant in this case is 1 over 5
So we could then write out our normalized state like this
Our probability of being in the spin-up state is
equal to the inner product between
Spin up and our normalized state squared
so that's going to work out to be  a  squared
Which is equal to 9 over 25
That's a probability of being measured in the spin up state.
Very similar for our spin-down probability, it would be the inner product between spin down and our normalized state squared.
Works out to be just the value of  b  squared
Which is this which is 16 over 25, and there's our probability of being spin down
So here's the solution to what you saw at the very start of the video
It's an example of the Schrodinger's cat thought experiment.
In this version a cat is stuck in a closed box with a poison vial and a
Stern-Gerlach analyzer with an input state of Chi the Greek letter
The one that looks like an X here. So this Chi input state is
described by
you know a
superposition of spin up and spin down.
The amplitude in front of both the spin up and spin down ket is this square root of 1/2
Now that's just been taken outside of the brackets, but it is the amplitude in front of both of those ket states.
So the release valve from the poison vial is
linked to the Stern-Gerlach analyzer and the valve will be opened if a
spin up Neutron is measured. Before you open the box,
the neutron is in a superposition of spin up and spin down and the cat is in a superposition of alive and dead.
We're asked, based on this input state, chi,
what is the probability that Schrodinger's cat is alive when we open the box
So based on what we know,
the probability of one of these states I guess occurring is
You're going to take the amplitude in front of it, and you're going to square that value
So if we're going to square the one in front of what we've got here
We're going to end up with a probability of 1/2. So
That's the correct solution. Here we are like this is exactly how you would do it mathematically.
Here, they're calculating the inner product between spin down and chi which is I guess the
probability that the poison vial is never opened so you can just do 1 minus whatever you're getting out of this answer to find out
the probability that the
Poison is opened
So they do that inner product and they write out chi explicitly as what we've been given to be its definition
you can take
everything out you, end up with a couple of inner products that are very familiar to us now
This one is 0 and this one is 1
Try and follow through with that math right there, you'll get a half, it makes sense
I guess intuitively and now it makes sense mathematically. By the end of filming this video
I feel like I've been in my room talking to myself about quantum mechanics for so long that I worry
It makes sense to no one except me
I guess that's my biggest fear putting this up is that it just makes no sense at all
But it made sense to me in my brain
But yeah try and let me know like,
if you could follow along with it and what I could do to change. If you want to follow along with the exact course
that I'm using and try out some of the interactive quiz questions for yourself, then it's all on brilliant.org. Brilliant are a
Supporter of my channel and you can go to brilliant.org/tibees to sign up for free
You can also get 20% off an annual premium subscription if you're one of the first 200 people to follow that link
Thanks to Brilliant for supporting my channel, and I do hope you find their content to be useful
It's definitely helping me to relearn some of these concepts
Again, thanks for watching and I'll see you in the next episode
I think there are maybe one or two more episodes of this math for quantum mechanics before we fully covered this
mathematical underpinning, and maybe we can move on to another topic after that. Totally open to suggestions of what that next topic would be
