- This episode was made
possible by Brilliant. Hi everyone, Jade here. If you read any article or watch any video about quantum mechanics, I challenge you to find one that doesn't mention the wave function. It's basically the bedrock
of modern quantum mechanics. But it's a tricky subject, and a lot of mystery still
surrounds this elusive guy. In this video, I'm going to explain in plain English exactly
what the wave function is, as well as answer the top
three most voted questions to my post. What is something that
you do not understand about the wave function? As this is quantum mechanics, you'll probably never
be 100% not confused, but hopefully by the end of this video you'll be slightly less confused. So what is the way function, exactly? Well, let's start with the word itself. Wave function. Wave function. As you probably know from high school, a function is a process
that takes in a value and spits out a related value. For example, a very simple
function is y equals x, where the output value is exactly the same as the input value. A slightly more complicated function is y equals x squared, where
the output value is the square of the input value. The shape this function
makes is called a parabola. Some functions described
real-world phenomena. For example, if you throw
an object in the air, it'll trace out the path
of an upside down parabola. Just like this function
describes the behavior of a throne object, the wave function describes the behavior of a quantum particle. So what is this behavior like? Well, let's look at the
other half of the word, wave. The behavior of a quantum
particle can always be described by a combination of our two
favorite waves from high school, the sine wave and the cosine wave. We'll go into more
detail on this in a bit, but for now the wave function
is a mathematical description of a quantum object, which
takes the form of a wave. Let's look at an example. This is a standard university
example of a wave function. It describes the behavior of a quantum particle, say
an electron trapped in a box which can only move in one dimension. So what does this wave function mean? Like the parabola, does
it trace out the path of the electron? No, it describes something
totally different. The x-axis describes where
the electron can be in space. But what about the y-axis? What does the value of the wave function here physically mean? Well, when we square the
value of the wave function, for reasons I'll explain later, this gives us something called
the probability amplitude. The probability amplitude can be thought of as a wave of probability. It describes the likelihood of finding the electron in a given place if we went to look for it there. The idea of a wave of probability
is a bit hard to grasp. So let's imagine a more
down-to-earth scenario. Imagine a jewel thief has
just been let out of prison. She didn't learn anything from her time and immediately starts
stealing from houses again. While they can't pinpoint
her exact location, the police can assign probabilities to where she'll strike next. For example, knowing the kind of property she's targeted in the past, the police can guess that
there's a higher probability she'll target the wealthy neighborhoods with their higher value
jewels than the poor ones, this can be thought of
as a wave of probability. It's not tangible, and it's not real, just to set of abstract
numbers that can be assigned to various parts of the city. In a similar way, a wave function describes
different probabilities to different locations of the box, based on how likely it is
we'll find our electron there. Now, what if the police managed to catch the thief red-handed? Immediately, their spread
out probability distribution collapses to being single
point at one location and not anywhere else. Likewise, if the electron is
detected in a certain location, it's a wave function
instantly collapses to a point and there is zero probability of finding it anywhere else here. But here, our analogy breaks down. The thief hasn't actually
spread herself out across the city. In reality, she's only ever
in one place at a time. The reason that the police
can't pinpoint her location is because of lack of information, not because of an actual
property about her. The electron on the other
hand is very different. When we don't know its position, we can't assume it's
in some definite place. It's not because of a lack of information that we can't pinpoint it. It's a fundamental property
about quantum objects. It's position really is
spread out like a wave, and it's only when we measure it does it collapse into a
particle with a single location. As I mentioned earlier, when it comes to quantum mechanics, you will most likely never be at least not a little bit confused. But let's get back to our example. So the probability amplitude, the y-axis, describes the probability
of finding our electron in a particular position. For example, there's a high probability we'll find our electron here, a low probability we'll find it here, and a zero probability,
we'll find it here. Here's what some other
way functions look like, and the exact same rules apply. When we square it, we get
the probability distribution of where the electron is likely
to be when we measure it. Now, about why it's squared, what I learned in university and what the classic textbook example is, is that the reason that we square it is because we don't wanna end up with a negative probability. Why don't we want to negative probability? Because negative
probabilities don't exist. They can't be a negative 40% chance that it will rain tomorrow or a negative 4% chance
you'll draw an ace. The lowest probability
you can have is zero. When you square a negative number it becomes a positive number. So that fixes that problem. So that's the wave function. Now as promised, I'm going to answer three
additional questions that were the most
highly voted on my post. What is something you do not understand about the wave function? Zach Heilman asks, "What's the connection
between the wave function and wave-particle duality?" This is a great question. So wave particle-duality is the phenomenon that quantum objects
sometimes act like waves, and sometimes like particles. We saw this in our box example where before the position of
the electron was measured, its probability was
spread out like a wave, but then as soon as we measured it, it collapsed into a particle. In other words, the
probability of a particle being in a location is a wave, but the actual physical appearance of the particle is not. A famous example of this is
the double-slit experiment where individual electrons
hit the screen as particles but added up, they make an
interference pattern like a wave. The wave functions simply
describes this behavior. The probability amplitude which
is spread out like a wave, predicts where the electron
will be measured as a particle. That's why when we fire
many electrons at a screen they make a typical wave
interference pattern. So I hope that answered your question. Now for the next one, Mayank Bhandare asked, "Why do we have it in the first place? Especially if it's
something that isn't real?" This is a very common question, but it's not universally agreed that the wave function isn't real. Many people think that it
is just a mathematical tool with no physical existence. But I think this table from Wikipedia, sums up the state of affairs pretty well. We can see the different
interpretations down the side, and what they disagree
about along the top. If we look at the real
wave function column, we can see that it's indeed very disputed, and even two of the most popular theories of quantum mechanics, the
Copenhagen interpretation and the Many-Worlds
interpretation do not agree. I was pretty confused about
this during my degree too. Interpretations of quantum physics weren't talked about very much and more of the focus was on the math and solving the equation. Most textbooks just treat
it as a mathematical tool. And the wave function collapse is seen more as the wave
function equation changing rather than a real physical wave changing. This is all you really need to
know for practical purposes, like figuring out the probability of where the electron is
and solving the equation. Some quantum physicists
go so far as to say that it's pointless to
even ask the question of what the wave function really is. I'm not super into this answer because I think that
conceptual understanding is really important. But at the moment we just
simply don't know what it is. And hey, if it makes you feel better, even Schrodinger didn't know
what his equation meant. He thought it represented electric charge. So to answer the question, why do we have it in the first place? Even if we assume it's not real, it's incredibly useful for predicting the behavior
of quantum objects. It's like our parabola. There isn't a real function in the air that the bowl is rolling along, but this piece of mathematics
describes the motion of this ball extremely well. Likewise, the wave function
describes the behavior of quantum particles extremely well. So we can use it to predict things and create cool devices
like quantum computers, transistors, lasers,
smartphones, GPS, and more. And finally, I saved the best for last. Master Adit asks, "How did Schrodinger
derive the wave equation?" Often seeing the inner workings and train of thought behind an equation can reveal a lot of the intuition. But if anything, this story reveals more about
Schrodinger than his equation. So basically Schrodinger was asked to give a talk on De Broglie
his idea that particles have wave-like properties. And after the talk, a
professor in the audience named Peter Debye was
like, hey, Schrodinger. If particles can like waves
there must be a wave equation. I want you to find it. And Schrodinger was like, okay. So he went to a secluded
mountain cabin to work. Schrodinger was a polygamist
and didn't take his wife but was surrounded by
beautiful young women. And in about two months he produced four papers outlining the infamous wave equation. But what's interesting is
if you look at the papers, there actually isn't a derivation. It kind of just jumps
right to the equation. Nobel prize, winning physicist,
Richard Feynman said of him, "When shredding a first
wrote down his equation, he gave a kind of derivation based on some heuristic arguments and some brilliant, intuitive guesses. Some of the arguments
he used were even false. But that does not matter. The only important thing is that the ultimate equation
gives a correct description of nature. He also said, "Where did we get that equation from? It's not possible to derive
it from anything you know. It came out of the mind of Schrodinger. So I'm sorry to say that in this case, knowing the origins of the equation, doesn't really shed much light on it. Schrodinger must have just
had an incredible intuition and inspiration. I think this story goes really nicely with the reputation of quantum mechanics. So much mystery and
intrigue around the topic that it seems fitting that its main equation came from a man just as mysterious and intriguing. If you do you have any more questions, the best way to familiarize
yourself with a topic is to work through problems and spend time forming an intuition. There's no way I could have
gotten such a deep feel for quantum mechanics if I didn't solve a few
way functions at uni. But you don't need to do a
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sign up using this link. Just go to brilliant.org/UpAndAtom. The link is in the description. Thank you for watching. Hopefully now you are
slightly less confused. Please let me know in the comments if you found this video helpful. As always, I'd like to thank my patrons for supporting this channel. These videos would not
be possible without you. I probably won't see you until next year. This is my last video of the year. So I wish you a safe and happy holiday, and I'll see you in 2020. Bye.
