Hey Crazies.
Schrodinger's equation governs the behavior of tiny particles.
Of course, they're not really particles.
They're actually tiny waves. Waves that sometimes give the illusion that they're particles.
But are they, though?
Are they waves?
Is Schrödinger's equation a wave equation?
Well, maybe not.
A viewer named Orden mentioned there were only three partial differential equations:
the wave equation, the heat equation, and Laplace’s equation.
At least for classical theoretical physics, anyway.
Now, with that many qualifiers, I'm not sure that statement says anything all that profound.
But their comment goes further.
Orden is suggesting that Schrödinger's equation isn't a wave equation at all, but is actually the heat equation applied to quantum particles.
And you're taking that seriously? Yeah, why not?
Okay, so what makes an equation a “wave equation”?
That sounds like a great place to start.
Before we get into the abstractness of quantum mechanics, let's look at something a little more tangible.
Consider a one dimensional string like this.
There's some tension in the string all along its length.
It's not perfectly rigid, though.
It can stretch if we push it up or down.
Naturally, this would increase the tension ever so slightly.
That little bit of extra tension is something very special.
It's called a restoring force.
It's the force that will restore the string to equilibrium once we let go.
Or at least it'll try.
The horizontal components of the tension always stay the same. They're just the equilibrium tension.
It's only this tiny vertical component that changes.
That's the restoring force, balanced only by the force from my finger.
And we all know from my prestigious laws of motion, particularly the second law, that any unbalanced force will cause an acceleration.
What he said.
So when we let go, that restoring force will accelerate the string toward equilibrium.
But by the time it gets there, it'll be moving.
That string has inertia, too. It's not going to stop just because it got to equilibrium.
It's going to keep going, causing a restoring force to grow in the opposite direction.
That new restoring force will slow the string until it ultimately stops and comes back again.
The string oscillates.
It won't settle at equilibrium until all the energy is released through heat, which doesn't actually take that long.
This all happens pretty fast. Fast fast!
I mean, here it is in real life. One pluck, a little vibration, and it's over.
Anyway, we'd like to have some equation that describes how all that works.
A wave equation, if you will.
I'm not going to go through a whole mathematical derivation because that's not what we do here at the Science Asylum.
But I've got some good links in the doobly-doo. You know, if you're interested.
We're going to look at this a bit more intuitively.
Written in its full form, the wave equation is, well, a lot to look at.
But this big fraction here, that's just the acceleration.
There’s one of those at every point along the string. So you might be wondering about this other fraction over here.
Well, that's ... what I'm trying to film a video here.
A memo from nerd clone?
Actually, derivatives aren't fractions.
Apparently, I'm not allowed to call derivatives fractions. Fine.
This ... stuff over here?
That's just concavity.
These sections of the string are concave up.
These are concave down.
Concavity just tells us which way the string is curving and by how much.
So curviness determines acceleration? Yep.
How? For that, we need to look a little closer.
Every point along the string has an acceleration and a measure of curviness.
Let's consider just a tiny piece of the string - a piece that only includes one of those points.
Each end has a tention tugging on it from the rest of the string.
Those tensions are the same because tension is uniform throughout the string.
But the components are not the same.
In fact, this piece has an unbalanced tension upward, which causes its upward acceleration.
Called it!
As Newton said earlier, unbalanced forces cause acceleration.
But this unbalanced tension depends on the curviness of the string, which means the acceleration also depends on the curviness.
That's exactly what the wave equation says.
This relationship shows up all over physics from strings to sound to light.
You just have to change this quantity here. That's the deviation from equilibrium.
For strings, it was a physical distance.
The same is true for sound.
For light, it's a deviation in the value of the electromagnetic field, not a distance.
But what about quantum mechanics?
Well, that stuff is described by Schrödinger's equation, and the solutions we extract from it are this - often called the wave function.
These curly d’s tell us there are changes in that wave function ...
changes both across space and over time.
Unfortunately, if we compare this with the wave equation, there's a bit of a problem.
These twos tell us the spatial changes are second order changes.
In other words, they're changes of changes.
The wave equation has one of those over time as well.
But Schrödinger's equation only involves a first order change in time.
That is not a wave equation.
Then why are the solutions called “wave functions”? Well, because language is complicated.
Language is a very fluid thing. It's constantly changing.
Some words are so old that they go out of fashion.
Others are brand new.
Science tends to have the opposite problem.
We only come up with new words when new science happens.
And then we keep those words forever.
Even if we eventually find out those words are terribly misleading or confusing.
Sometimes it isn't even scientists that come up with them.
A lot of the time it's journalists. But that's a rant for another day.
Anyway, the word “wave” has a lot of different definitions depending on context.
One of the earliest is the verb, meaning moving your hand back and forth.
Its earliest use as a noun was in reference to a moving billow of water.
These days, the word wave is used to describe anything that undulates or moves in a repeating pattern.
Basically, anything that fits a pattern like one of these.
That's pretty much anything involving a sine or a cosine.
Are those solutions to the wave equation? Sure.
But they’re also solutions to a bunch of other equations.
You mean like that of a harmonic oscillator, Nick? Arvin Ash?
For those of you who may not have heard of it, a classical harmonic oscillator describes the motion of a spring.
Now, this may seem trivial, but it turns out that the math of the motion of a simple spring, if quantized, can describe nearly everything
we can see in the universe, because its math is almost identical to the math of the prevailing theory of reality: Quantum Field Theory.
How is a simple spring related to everything?
Check out my video on the Arvin Ash channel after you’ve viewed Nick’s video, of course.
Huh. Cool.
You should go check out Arvin’s video when you're done here. Where was I?
Oh, right. Sines and cosines.
They're common solutions to all sorts of differential equations, including Schrödinger's equation.
And we certainly call these shapes waves.
But that's not enough to make Schrödinger's equation a wave equation.
Wave equations have second order changes in both space and time.
Schrödinger's equation has second order changes in space, but only first order changes in time.
So what is it then? It's a heat equation.
I've done a video about this before, but here's the quickie version.
The heat equation relates changes in temperature over time at a particular location
to differences in temperature between nearby locations.
The bigger the difference, the faster the change and vice versa.
It tells us how we'd expect heat to flow inside of a solid body.
Unbelievable.
But just like the wave equation, it's not limited to its original purpose.
This quantity doesn't have to be temperature. It could really be anything that flows.
And did you notice this? It's a first order change in time. Just like Schrödinger's equation.
What flows in quantum mechanics? Probability.
I mean, technically, probability density ... but that's not a nuance that's important here.
Anyway, here's the 3s orbital in hydrogen atom.
The electron has a probability of being at any location on this map, but it's more likely to be in a brighter spot than a darker spot.
And that probability must be conserved, just like the mass in a fluid flow or the energy in a heat transfer.
Any change in probability at one location results in changes at other locations.
Probability flows through space and ultimately the electron finds itself in another quantum state.
This is the kind of process that the heat equation describes, but we still have one minor discrepancy.
Schrödinger's equation has an extra term that the standard heat equation doesn't.
The keyword there being “standard.” Heat equations can have extra terms.
For example, if there is a moving fluid, we account for that with a convection term.
But you'll notice the extra term in Schrödinger's equation doesn't involve changes at all.
It represents an independent external influence, like the nucleus of an atom.
The electron in a hydrogen atom is near a proton. That's what makes it hydrogen.
So this extra term accounts for that interaction.
It's an interaction term. We add them to the heat equation all the time, it's not that unusual.
In fluid dynamics they represent some kind of source or sink.
In Schrödinger's equation it's a source of energy otherwise unaccounted for because Schrödinger's equation is derived from conservation of energy.
Conservation of energy shall not be violated!
It's just the heat equation with an extra term.
So yeah, Orden, that's exactly what it is. And that kind of makes sense when you think about it.
There's nothing actually waving in this electron orbital.
I mean, sure, you could graph out the pattern and the shape is wavy, but there's no actual waving taking place.
It's a static graph.
The “wave equation” isn't about wave shapes.
It's about wave motion.
Remember, it was originally derived to explain motion like this: waves on a string.
The wave equation relates concavity to acceleration, curviness to oscillation.
That's not what's happening in quantum mechanics.
Schrödinger's equation doesn't relate curviness to oscillation.
It relates curviness to flow.
Specifically the flow of probability.
Probability flows from one place to another as time passes.
That is what Schrödinger's equation describes.
To be fair, though, and I've said this in previous videos, Schrödinger's equation is not a complete description of quantum phenomena.
You know all these solutions we have for the hydrogen atom?
Well, we had to make several big assumptions to get there.
Essentially, we've ignored what we call the fine structure of the atom.
In case you didn't know, I have a book. It's available as an e-book and in paperback.
The e-book is also available as a digital download on Patreon if you pledge more than $5 per month.
Shameless self-promotion: Check! Links in the doobly-doo.
Anyway, there are actually better options than Schrödinger's equation.
The Pauli-Schrödinger equation allows for quantum spin and magnetic fields.
If the particles are relativistic, we've got the ever popular Dirac equation.
If you still really want it to be a wave equation though, there's the Klein-Gordon equation.
It's got second order changes in both space and time.
But the main takeaway here is not to get too caught up on the language.
It's just going to leave you stressed - it's not worth it.
The relationships are what's important, not the names.
That being said, don't forget to check out Arvin’s video on harmonic oscillators.
He makes really good stuff.
And until next time, remember: it's okay to be a little crazy.
Stroheim333 pointed out that my Earth-centered solar system breaks some physical laws.
Well, not exactly. It's just that the rules you're familiar with are written from an inertial perspective.
And my Earth-centered system was not inertial. Anyway, thanks for watching.