- [Instructor] So when people first showed

that matter particles like electrons

can have wavelengths and
when DeBroglie showed

that the wavelength is Planck's constant

over the momentum, people were like cool,

it's pretty sweet.

But you know someone
was like wait a minute,

if this particle has wavelike properties

and it has a wavelength,

what exactly is waving?

What is this wave we're
even talking about?

Conceptually it's a little strange.

I mean a water wave, we know what that is.

It's a bunch of water that's
oscillating up and down.

A wave on a string, we know what that is.

This string itself is moving up and down

and it extends through space.

But it's hard to imagine,

how is this electron having a wavelength

and what is the actual wave itself?

So physicists were
grappling with this issue,

trying to conceptually understand

how to describe the wave of the electron.

They wanted to do two things.

They wanted a mathematical description

for the shape of that wave,

and that's called the wave function.

So this wave function gives
you a mathematical description

for what the shape of the wave is.

So different electron
systems are gonna have

different wave functions,

and this is psi,

it's the symbol for the wave function.

So this is psi, the psi symbol.

It's a function of x.

So at different points in x,

it may have a large value,
it may have a small value.

This function would give
you the mathematical shape

of this wave.

So that was one of the things
they were trying to determine.

But they also wanted to interpret it.

Like what does this
wave function even mean?

So we've got two problems.

We want a mathematical
description of the wave

and we wanna interpret what
does this wave even mean.

Now the person that gave us
the mathematical description

of this wave function
was Erwin Schrodinger.

So Schrodinger is this guy right here.

Schrodinger's right here.

He wrote down Schrodinger's Equation,

and his name now is basically synonymous

with quantum mechanics
because this is arguably

the most important equation
in all of quantum mechanics.

There's a bunch of partial
derivatives in here

and Planck's constants,

but the important thing is that it's got

the wave function in here.

Now if you've never
seen partial derivatives

or calculus, it's okay.

All you need to know for our
purposes today in this video

is that this equation
is a way to crank out

the mathematical wave function.

What is this function that
gives us the shape of the wave

as a function of x?

And you could imagine
plotting this on some graph.

So once you solve for this
psi as a function of x,

you could plot what this looks like.

Maybe it looks something like this,

and who knows, it could
do all kinds of stuff.

Maybe it looks like that.

But Schrodinger's Equation is the way

you can get this wave function.

So Schrodinger gave us a way to get

the mathematical wave function,

but we also wanted to interpret it.

What does this even mean?

To say that this wave function
represents the electrons

is still strange.

What does that mean?

Schrodinger tried to
interpret it this way.

He said, okay maybe this electron
really is like smudged out

in space and its charge
is kinda distributed

in different places.

Schrodinger wanted to
interpret this wave function

as charge density,

and I mean it's kind of
a reasonable thing to do.

The way you get a water
wave is by having water

spread out through space.

So maybe the way you get an electron wave

is to have the charge of the electron

spread out through space.

But this description didn't work so well,

which is kinda strange.

Schrodinger invented this equation.

He came up with this equation,

but he couldn't even interpret

what he was describing correctly.

It took someone else.

It took a guy named Max Born to give us

the interpretation we go with
now for this wave function.

Max Born said no, don't interpret
it as the charge density.

What you should do is interpret this psi

is giving you a way to get the probability

of finding the electron
at a given point in space.

So Max Born said this,

if you find your psi,
like he said go ahead

and use Schrodinger's
equation, use it, get psi.

Once you have psi, what you do
is you square this function.

So take the absolute value, square it,

and what that's gonna give
you is the probability

of finding the electron at a given point.

Now technically it's
the probability density,

but for our purposes,

you can pretty much just think about this

as the probability of finding the electron

at a given point.

So if this was our wave
function in other words,

Max Born would tell us that
points where it's zero,

these points right here
where the value is zero,

there is a zero percent
chance you're gonna find

the electron there.

Points where there's a large value of psi,

be it positive or negative,

there's gonna be a large probability

of finding the electron at that point.

And we could say the odds
of finding the electron

at a given point here are gonna be largest

for this value of x right here

because that's the point
for which the wave function

has the greatest magnitude.

But you won't necessarily
find the electron there.

If you repeat this
experiment over and over,

you may find the electron here once,

you may find it over here, you
may find it there next time.

You have to keep taking measurements,

and if you keep taking measurements,

you'll get this
distribution where you find

a lot of 'em here, a lot of 'em there,

a lot of 'em here, and a lot of 'em here,

always where there's these
peaks you get more of them

than you would have at other points

where the values are smaller.

You build up a distribution
that's represented

by this wave function.

So the wave function does not tell you

where the electron's gonna be.

It just gives you the probability,

and technically the square of it

gives you the probability of
finding the electron somewhere.

So even at points down here
where the wave function

has a negative value,

I mean you can't have
a negative probability.

You square that value.

That gives you the probability
of finding the electron

in that region.

So in other words,

let's get rid of all this.

Let's say we solved some
Schrodinger equation

or we were just handed a wave function

and we were told it looks
like this and we were asked,

where are you most likely
to find the electron?

Well the value of the
wave function is greatest

at this point here,

so you'd be most likely
to find the electron

in this region right here.

You'd have no shot of
finding it right there.

You'd have pretty good odds
of finding it right here

or right here,

but you'd have the greatest
chance of finding it

in this region right here.

So you'd have to repeat
this measurement many times.

In quantum mechanics,

one measurement doesn't
verify that you've got

the right wave function.

Because if I do one experiment

and measure one electron,

boop I might find the
electron right there.

That doesn't really tell me anything.

I have to repeat this
over and over to make sure

the relative frequency of
where I'm finding electrons

matches the wave function
I'm using to model

that electron system.

So that's what the wave function is.

That's what it can do for you,

although if I were you,

I'd still be unsatisfied.

I'd be like, wait a minute,
okay, that's fine and good.

Wave function can give us the probability

or the probability density
of finding the electron

in a given region,

but we haven't answered the question,

what is waving here and what exactly

is this wave function?

Is this a physical object
sort of like a water wave

or even an electromagnetic wave?

Or is this just some mathematical trickery

that we're using that has
no physical interpretation

other than giving us
information about where

the electron's gonna be?

And I've got good news and bad news.

The bad news is that
people still don't agree

on how to interpret this wave function.

Yes they know that the
square of it gives you

the probability of finding
the electron in some region,

but people differ on how
they're supposed to interpret it

past that point.

For instance, is this wave function

the wave function of a single electron

or is this wave function
really the wave function

of a system, an ensemble of electrons,

all similarly prepared
that you're gonna do

the experiment on?

In other words, does it
describe one electron

or only describe a system of electrons?

Does it not describe the electron at all

but only our measurement of the electron?

And what happens to this wave function

when you actually measure the electron?

When you measure the electron
you find it somewhere,

and at that moment there's no chance

of finding it over here at all.

So does the act of measuring the electron

cause some catastrophic
collapse in this wave function

that's not described by
Schrodinger's Equation?

These and many more
questions are still debated

and not completely understood.

That's the bad news.

The good news is that we don't really need

to understand that to make progress.

Everyone knows how to
use the wave function

to get the probabilities of measurements.

You can have your favorite interpretation,

but luckily pretty much
regardless of how you interpret

this wave function,

as long as you're using it correctly

to get the probabilities of measurements,

you can continue making progress,

testing different models,

and correlating data to the measurements

that people make in the lab.

Now I'm not saying that interpretations

of this wave function are not important.

People have tried cracking
this nut for over 100 years,

and it's resisted.

Maybe that's because it's a waste of time

or maybe it's because the
difficulty of figuring this out

is so great that whoever
does it will go down

in history as one of the
great physicists of all time.

It's hard to tell right now,

but what's undebatable is
for about 100 years now,

we've been able to make
progress with quantum mechanics

even though we differ on
how exactly to interpret

what this wave function really represents.

So recapping the wave function gives you

the probability of finding a particle

in that region of space,

specifically the square
of the wave function

gives you the probability density

of finding a particle
at that point in space.

This almost everyone has agreed upon.

Whether the wave function
has deeper implications

besides this, people differ,

but that hasn't yet
stopped us from applying

quantum mechanics correctly in a variety

of different scenarios.