Hey Crazies. Ready for some more quantum mechanics?
Yeah, me neither. But I’m crazy, so I’m going do it anyway.
In the previous video, I mentioned the periodic table a few times and said that quantum mechanics
perfectly predicts it. This, of course, left Borg972, Blazej Michalik,
and others wondering, you know: How’s it do that?
Well… Samarth Sai was asking about this thing.
That’s the hydrogen atom’s wave-function and…
…before we can understand the periodic table… …we need to have some idea of what wave-functions say.
YOU CAN HANDLE THIS… I PROMISE!
What’s a wave-function? Yeah, that seems like a decent place to start.
A Wave-Function is just a mathematical representation… of a particle.
It contains everything you can possibly know about a particle’s properties.
Yes, there is only (finger) one single wave-function for a particle in given state.
All three of these graphs from the previous video are found using the same wave-function,
just written in terms of different measurements.
In fact, you can write that same wave-function in terms of whatever measurement you want
to make a prediction about. All you need is a Fourier Transform.
They’re a total pain in the butt, but they work.
Anyway! As you can see, inside of an atom, position
isn’t very useful. There are a lot of possible locations for
the electron. If you want useful information, then it’s
best to stick to definite measurements like energy.
We call those the “stationary states” of the particle because they’re stable and
steady. But those contain more information than just
energy. Since a particle’s Energy and part of its
angular momentum are definite at the same time, they can be measured at the same time.
That means the same stationary states work for both.
It’s super convenient!
Which brings me to something Jade from Up & Atom and Josh Webb were asking about.
What’s the deal with orbitals? Orbitals are, well, they’re very poorly
named. Yes, the electrons inside atoms are moving.
They have both linear momentum and angular momentum but that doesn’t mean they’re
orbiting. If they were, those orbits would collapse.
Remember, electrons don’t obey the rules we’re used to.
You must unlearn what you have learned. Just think of orbitals as subdivisions of
an energy level. The size of a city block can determine how
many subdivisions it will have. A small block might have one subdivision where
as a large block might have four. The same thing happens in atoms, but it depends
on how much energy is available.
Like usual, let’s number the energy levels: 1, 2, 3, 4, 5, etc.
Sometimes energy levels are called “shells,” so these are Shell numbers.
Those numbers also tell us how many orbital types there are.
Shell number 1 contains 1 orbital type, but shell number 4 contains 4 different types.
You can label these types with another number based on angular momentum or with a letter
for historical reasons we won’t go into. Each of those types is then subdivided into
a number of orbitals based on how the angular momentum is tilted.
And before someone comments “You forgot to mention the shapes!”
You know who you are, Ben! Yes, different orbitals have different shapes
and orientations, but that doesn’t really matter until atoms start bonding to each other.
Forget about that for right now. All that matters today is how many orbitals
there are in a shell. That’s what tells us how much room the shell
has for electrons, which is why different rows on the periodic table have different
numbers of elements.
But isn’t each of these sections twice as big as it should be?
Yeah… that’s where spin comes in. Particles seem to have a little more angular
momentum than they should. Just a smidge, but it’s measurably there.
When we were trying to name it, of course we resorted to an analogy.
The Earth has orbital angular momentum because it orbits the Sun. But it also rotates, so
it has a separate spin angular momentum. So electrons are spinning inside atoms as
they orbit? No… No, they don’t do anything like that.
Electrons do not orbit. Electrons do not spin. We know not where they’re going. We know
not where they’ve been.
In quantum mechanics, the term “angular momentum” implies very little about motion,
so I’m just going to keep the word “spin” in quotes from now on.
Whatever “spin” is, it’s an intrinsic property of a particle. It can’t ever change.
Electrons, protons, and neutrons are all Spin 1/2 Particles,
meaning their angular momentum is always off by half a Planck constant either one way
or the other depending on orientation. Why is this important?
Oh, right, right. Any particle with one of these spins is called
a fermion, named after the epic historical physicist Enrico Fermi who you might remember
from the orders of magnitude video. Fermions have a problem. They don’t get
along very well. You can’t ever have two in exactly the same
state at exactly the same time.
You can only have two in each orbital, which forces some of the electrons into higher
energy states. Shouldn’t that be one electron per orbital?
Oh no, my friend. Two. Like I just said:
You can’t ever have two in exactly the same state at exactly the same time.
That means if at least one of the quantum numbers is different, it’s totally fine.
A specific orbital is labeled with 3 of those numbers.
Spin is also a quantum number and the spin orientation another.
Even though these two electrons are in the exact same orbital, their spins are opposite,
so it’s allowed.
Those two spin states are the reason the sections of the periodic table are twice as big as
the number of orbitals. s-orbitals can hold 2, p-orbitals can hold
6, d-orbitals can hold 10, f-orbitals can hold 14, and so on.
However, you’ve probably noticed row width doesn’t grow for every row of the table.
That’s because energy shells don’t have clear boundaries.
They’re not separate things like in a Russian nesting doll.
They overlap a little, so the orbitals inside them don’t get filled in the order you’d
expect. Orbitals are filled from lower to higher energy,
which isn’t necessarily in shell-number order.
Just like with anything real, it’s complicated. Luckily, as atoms get bigger, the types of
orbitals stay the same. It’s only the energies that change
and, since electrons are the foundation of chemistry, the repeating patterns will group
similar chemical elements into columns. It doesn’t have to be rectangular though.
It could just as easily be circular as long as the groups line up.
Rectangles are just easier to read. So, what shape do you think the Periodic Table
should be? Let me know down in the comments and I’ll
read them… periodically. Thanks for liking and sharing this video.
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And until next time, remember, it’s OK to be a little crazy.
The featured comment comes from Raju who pointed out the concept of atoms was being discussed
in India before Ancient Greece. This is technically true. I admit it was a
bit euro-centric of me, but the further back in time you look the more religious and the
less scientific those discussions were. It’s difficult to know where to draw the
line.