I'm pretty bad at quantum mechanics.
I talk a lot about physics here on my channel and I think you guys often credit me
with being more of an expert than I really am.
I am doing a PhD in physics
but it's in an area of biophysics and I learn a lot more now about the physics of how plants grow than I do about
quantum mechanics and the physics of how particles behave on a very small scale.
But in saying that, quantum mechanics I still find to be an inherently very interesting field.
I sort of think that most people who end up being interested in physics in some way
come into this interest through either quantum mechanics and the strangeness they've heard about at that scale or through something like
astrophysics and the vastness and the mysteries out there, things like dark matter and dark energy.
So I think quantum physics is a topic that a lot of sort of
hobbyist physicists like to learn.
People that are not necessarily trained in physics often like to read ideas about quantum physics, but
one barrier to really understanding some of quantum physics is the mathematics behind it and
quantum physics is a math heavy area. You simply can't
really understand everything that's out there and in books and online
without being able to understand all the mathematical underpinnings and frankly it's pretty complicated.
So even though I've passed multiple quantum physics courses along my physics education like through my degree
I wouldn't call myself an expert on it. And in fact, I've forgotten a lot of the
mathematical underpinnings that go with it and
because
pretty much the nature of how I learned quantum mechanics at uni was
studying and cramming for exams and just trying to get out assignments. And
i've sort of spoken before about how I don't like this approach to learning and how a lot of what I felt like I
learned during my undergrad degree I've almost forgotten or I never really learned well because I was just
memorizing a lot of information,
memorizing a lot of these mathematical techniques, being able to use them in the exams.
I was pretty good at being able to like  regurgitate them in that way
but there's still a lot more that I want to deeply understand about physics, about quantum physics.
So I thought I'd take you on a little bit of a journey with me. I'm going to
study and revise some quantum mechanics and specifically the
mathematical underpinnings, so I'm going to
basically build up all the math skills that you need to understand quantum mechanics
and I want to bring you along with me.
So this is hopefully going to be a bit of a series where we can learn together in a way so
whilst I might have more of a
physics and math
starting point, I hope that we can sort of learn together through this.
It's going to be a little bit of an experiment to see what level you guys enjoy me going at,
I know some people that watch my videos have like a degree or a PhD in physics already,
some people just have a mild interest in the subject.
So what I'm going to do is work through some online courses in quantum mechanics
and I'm going to look at the math behind it. So I'm going to move reasonably slowly and look in this episode at
pretty much
linear algebra and how that applies to quantum mechanics. So how you
write out
quantum states and how you do that in bra-ket notation.
This is
really often used in quantum mechanics, but it can be one of the most off-putting
things to see for someone who doesn't understand the notation.
I think it comes down to be reasonably simple.
And yeah the reason I want to relearn some of this mathematical notation is because I know how important it is to
understanding further ideas, and I don't think I understand it as well as I want to. Okay
so I'm working through a course called quantum objects on brilliant.org/Tibees, and I'm on their section for mathematical foundations.
I'll put a link down below so that you can find this course as well and study along with me if you like.
What they've spoken about so far is an experiment in quantum mechanics called the Stern-Gerlach experiment,
I hope I said that right. It's a pretty important experiment in quantum mechanics,
we'll talk about that experiment later, but first of all we're going to
touch on something that at first doesn't seem like it's related to quantum mechanics at all, but actually underpins a lot of the math
that's going on here and
it kind of touches on the essence of something called linear algebra. It's talking about basically
wavelengths of light and how different wavelengths of light give you different colors and so every color of the rainbow
can be described
pretty simply as a scalar value.
That's just meaning that every color on here corresponds to some wavelength in nanometers,
so 600 nanometers yellow.
So then were talking about
actually the really famous physicist James Clark Maxwell who you probably know from Maxwell's equations and
electromagnetism. He was a I guess really influential scientist and made a lot of contributions to physics.
So it's talking about something that he did
where back when they only had black and white photography he was trying to
get some color into photographs and sort of reconstruct some color.
So
let's see
Maxwell
captured black and white photos of the same colorful scottish tartan using red, green and blue filters
that only let one color of light through. So we had
projections of these red, green and blue images and
he combined these projections to try and recreate the color photo.
And what did it look like when he
superimposed the projections?
Well
in fact
he did produce an image with all the true colors intact so we can see he's made the original like tartan colors.
So apparently Maxwell was amazed to see that any color of the rainbow could be recreated in this
superimposed projection. Not just the red, green and blue colors that he had hoped for but what was amazing is that
like yellow and purple are showing up in here even though no yellow or purple wavelengths of light
were passing through
these filters. There's no purple or yellow wavelengths of light in these three images that he was using.
So yeah here what it demonstrates is that
combining red and blue light gives you purple light which appears purple to our eye
exactly the same way that
purely purple light would appear. What this is representing is that a
combination of colors can be represented as a linear combination.
So that's what we have written here. It's our first glimpse at the sort of notation that we're going to learn.
It's saying that
purple light
can be created from an equal mixture of
half red light and half blue light so that's what we've got going on here. So you can see that
the way they've written purple light, red light and blue light is in these sort of funny brackets,
and those are called ket brackets. Here
I think you just need to know that they represent a state so
the state of the light in the first set of brackets is purple. the state of the light here is red or blue,
It's just telling you, what color the light is. In this description here
the colors red and blue were serving as a color basis
which we combined in linear combinations to create new colors. So this thing here is the linear combination of
red and blue with equal parts of each. So then it's asking how many
basis
wavelengths or colors do we need such that we can combine them to produce every color?
It's saying well if we just has one color as our basis like blue
then all we could do was create different shades of blue. So if we are only using blue light in our linear combinations
we could never create any other colors.
We're going to need at least two colors to create something new. And the answer is that we need
three colors to create a basis to
create every other color of light. I think this has to do with actually like the biology of the human eye.
So it's a side effect of how our eyes perceive light.
Human eyes detect light using three different types of cell, each sensitive to long, medium and short wavelengths
which correspond to red, green and blue light. So these are actually the three colors that are gonna form
our basis in this example. So using our
building blocks of red, green and blue we can make any color such as
cyan, yellow, magenta.
They're all equal combinations of the red, green and blue
primary building blocks.
So in this example
there's a half in front of all the brackets meaning that all these mixtures are like equal parts of like
these two building blocks
but there would be like a
continuous series of colors that exist where
this half could be any real number so you can sort of have an infinite combination of
the building blocks to create an infinite number of new colors. There is a question here,
is it possible to construct the color blue through a combination of the colors, red and green?
The answer here is no because blue is one of our fundamental building blocks and we can't use the other building blocks
to make the third one.
We sort of need these three building blocks to start with to make any other further color.
This is called the color space and you can
represent it in this diagram here. So
they're saying
the three primary colors are shown as orthogonal axes since they are mutually exclusive.
A purely red source of light has no blue or green components.
It is impossible to represent one primary color as a linear combination of the others,
just as it's impossible to represent a coordinate on the z axis as a combination of x and y vectors.
So
here we're starting to move from our color example to
just like representing it on a grid. Like we've got x, y and z
axes and this whole thing about
orthogonal axes means that they're at 90 degrees to each other.
It just means that the green is one of our building blocks and it's going in a different direction to the blue building block and
they're going in a different direction to the red building block,
and you need all three of these to create a three-dimensional cube
essentially of color. When you add two vectors, say the red one and the blue one
and you end up coming out in this diagonal direction, that's where you get like a pink-purple type color.
So you've added the red and blue vector and got something in between. There's then a brief discussion about
normalizing vectors, and this means we're essentially
limiting the length of our vector. It's kind of like the intensity of the light that you want out at the end,
you want the intensity to be constant,
some value. So if you
crank up and down the red, blue and green values, you want them to
even out such that they always add to the same intensity. Typically for vectors
you normalize them all to be like
the unit length, or one, so if we want the intensity to be one and we're adding together two
base colors to get that intensity we'd take a half of
each color and there's a little discussion here about how you do that and make sure they do add up to something consistent.
And I think that this example here what the colors really does give us quite an intuitive idea of
What adding together different states means in terms of combining to give a new state.
Okay, so let's talk a little bit more about these brackets,
they're also called Dirac notation
because they I think were invented by Paul Dirac, a physicist and
so the one that we've seen here is the ket.
This time it's not containing a color but instead this Greek symbol psi,
and in the case of quantum mechanics the ket contains everything we know about the state of a quantum
object.
So what kind of information can it contain?
Well a ket must contain the information necessary to make physical predictions about the behavior of the state.
Essentially just think of that as like
what is the color of the particle?
What is something inherent about it? The example that we're going to go on to use is that we're going to have
this bracket represent whether a particle is spin up or spin down.
Now spin is a term used in quantum mechanics, and I read a little bit about it on
Brilliant on a previous course. Before we get to this mathematical course, there's a bit of an intro to the Stern-Gerlach experiment
where neutrons which are
neutral particles, they're one of the three subatomic
particles in an atom, so we've got protons, neutrons and electrons.
The neutrons don't have a charge and
so when you fire them through like a series of magnets you wouldn't expect them to be deflected
but we do find in this experiment that neutrons are
deflected as if they're acting like they have a charge or as if they're acting like tiny magnets
being attracted or repelled from the magnets that they're passing through.
So just
to sort of explain why these neutrons acted like tiny magnets
we said that they have a spin and
the spin doesn't mean that the neutrons are actually spinning. It I
believe was just termed to be spin because
objects that do spin often act like magnets and we're observing these particles act like magnets
so we say that they have a
inherent spin.
When you fire the neutrons through this experiment, through these magnets,
what you get on the other end is sort of two bands of the neutrons.
Some that were
deflected upwards a certain amount and some that we were deflected down a certain amount. The ones that went up we term they have a
spin up property and the ones that went down we say they are spin down.
I guess that's all you need to know about the spin at this point,
we just want to understand the maths of how you talk about this
so just know that we're going to have either an up arrow or down arrow in our ket
notation, and it just means what spin state is our particle in.
So here's a bit of a picture of this experiment
I was talking about, we put in this
particle where we just say psi, we don't know anything about the state of the particle, I guess.
And then it either becomes spin up state or spin down state.
These two states are
mutually exclusive
and
distinguishable, so
if you observe a neutron, the particle, in the spin up,
there is no chance that you will later observe it in the spin down state.
So we keep that in mind because when we want to find a mathematical representation
for our states we want it to share these properties
that states can be physically distinct and mutually exclusive. This takes us back to vectors and
working on a grid of 3D axes and x, y and z. So
orthogonal vectors, which is what these three are, x, y, and z are
mutually exclusive
because if you're pointing in the x
direction you can't be pointing at all in the z direction. That's sort of the way I understand it.
So this is all to say that
spin states can also be expressed as vector states too. So it mentioned in the last chapter that
spin states always seem to be some mixed combination of spin up and spin down
when we measure them.
So here's what I'm saying, that the state of a quantum particle
can be described as a linear combination of
some amount of spin up and some amount of spin down. The a and b here are just numbers
that represent how much of the spin up and how much of the spin down are combining to give us our final state.
So that's like saying how much of each color
goes into creating the final color if we were still talking about our previous example.
We only have a basis of two things here, like before we had three colors as our building blocks,
here our building blocks are just spin up and spin down. We don't need a third
building block to be able to describe our spin state.
So I'm saying we only need a basis consisting of two things here, but we can check it sort of mathematically.
So any basis set of vectors must possess three important mathematical properties,
normalization,
orthogonality and completeness.
So we can go on to see what these are about.
Now one term that I haven't used yet is a dot product or an inner product between two kets,
that's our state, reveals the amount of overlap between the two kets. So the state projected on itself
can be written as this, where we have the dot to represent the dot product, or
you flip around the first one to write it like this and it's just saying how
similar or what is the overlap between
these two states. In this case they're the same so there will be complete overlap and the answer will be one.
This is kind of visually what we're saying. We had these two vectors in space, they're exactly the same,
what is the overlap between them? Well, what is the inner product? It is one. If the
kets have this property then they are normalized and
that's sort of what we were talking about before is that they add together to give one and you can never have more than that.
So if we were still in our color space example, and we did the inner product of two red states,
we will get one because they're exactly the same. Now we want to check that our basis is orthogonal,
so if we have two different basis states,
that there is no overlap between them.
So in this example if we do the inner product between
the up state and the down state we find that it is zero.
That means
that they're essentially at ninety degrees and they're going in different directions.
There's no redundancy I guess in our basis states that we've chosen to use, our up and and our down.
Again, if we were to do this on our color example, and we
tried to see if
red and blue were orthogonal, we would get zero, we would find that they are orthogonal.
There's no similarity between them and they can be used as two
elements of our basis. These inner products can tell us if two states are orthogonal
but they can also tell you or give you an idea of how similar two states are.
So in this example here
we're considering these three colors, a pink, a purple and an orange and we're being asked
which of these two colors has an inner product closest to unity? So they're saying
unity means one
so we're really being asked which two colors are the most similar, and the answer is the pink and the purple. Just
visually looking at those they seem the most similar to our eye and so
you can even prove it to yourself here that if you did the inner product of
these two colors they would have give the answer that is closest to one. For our color space example
any of our colors could be made using some combination of red, green and blue and
for our quantum spin state it is complete because
any spin state can be represented as a combination of the kets up or down.
And this leaves us with the properties of what is called an orthonormal vector space and that's essentially what we need,
it's a linear algebra term. It's what we need to create a basis
and it's going to be really useful for
describing the outcome of like quantum mechanics experiments.
So our basis here that we formed is our up and down spin states. We've seen that they are normalized,
orthogonal and complete so let's try and do an example here.
Using these properties of the spin state
find the inner product of the up spin state on an unknown state vector.
We're doing the inner product and remember that's just checking how much overlap there is between
two things. We're doing the overlap between an up state and this psi function
which, it's definition is actually up here, It's some unknown
combination of up and down. I'll do this on paper...
So our general state vector, the psi, is given as a linear combination of our two orthogonal
basis kets, that's the up and the down state.
So we take this backwards ket here and times it by everything in brackets.
Really we take what's here at the front and times it by each term of the brackets and
we can take out the a and the b because they are scalars. They're just numbers and can be moved about freely.
We can simplify it based on what we know about the inner products of these
spin states already, that if they're the same with complete overlap, it's 1 and that if they're the opposite
it's going to be 0. That means our final answer just works out to be a.
Question here on the next page. It's doing the inner product of
two psi states, remember we do have the definition of that so try to work through that yourself,
the answer is pretty easy to check but see if you can get it out.
If you have managed to follow this far, you're actually doing really well and you're well on your way to be able to
understand a lot of
experimental
interpretations of quantum mechanics because this whole thing of taking two wave functions, which is what these psi's represent and
being able to do algebra with them is
really most of the maths that you're going to need to be able to do. You're going to need to see like
if you compare wave functions between particles, you take inner products what sort of results are you going to get?
If you did get to this point, I think that's awesome.
If you were able to,
you know, work through this yourself and actually understand the linear algebra up to this point and how it can be useful
for
constructing a basis and how you construct a basis,
and how we can do that in quantum mechanics, not just with colors and not just with vectors
but actually with something as abstract as a state, then well done to you.
I was really hazy on all of this until I actually read through this material on brilliant
just like sort of before I filmed this video.
So even though I'm here sort of telling you what I know as if I am the expert it's kind of what I've just
learned immediately before filming. So if there's a mistake or ways that I can better explain things,
let me know and I'm really excited to keep going
and to keep trying some more of these problems.
But I really want to know what you guys think and I want you to come along with me and and help me
define what this learning is going to be and what you actually want me to talk about.
If you do want to work through this course with me
then you can go to brilliant.org/Tibees and sign up for free,
there's also an annual premium subscription and the first 200 people that go to that link
can get 20% off. I think that this particular course on brilliant is really good because
it makes your learning active by giving you fun and challenging problems to try along the way.
So do check it out if you're interested, and thank you so much for watching.
I'd love for you guys to post questions down below and I really want to improve the way I explain these ideas
so if you again like have
suggestions for me or things you'd want me to explain, please let me know.
