so when people first showed that matter
particles like electrons can have
wavelengths and when dubrow's 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 wave-like
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
the 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 going to have
different wave functions and this is
this is psy it's the symbol for the wave
function so this is psy the size symbol
it's a function of X so at different
points in X it may have a large value
and may have a small value this function
would give you the mathematical shape of
this wave so I was one of the things
they were trying to determine but they
also wanted to interpret it like what is
this wave function even mean so we've
got two problems we want a mathematical
description of the wave and we want to
interpret what is this wave even mean
now the person that gave us the
mathematical description of this wave
function was Irwin Schrodinger so
Schrodinger is this guy right here
Schrodinger's right here
he wrote down Schrodinger's equation and
his
aim 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 can imagine
plotting this on some graph so once you
solve for this sigh 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 maybe this
electron really is like smudged out in
space and it's charge is kind of
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
kind of 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 nah don't interpret it as the
charge density what you should do is
interpret this sy 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 sigh
like he said go ahead and use
Schrodinger's equation use it
get the sigh once you have
what you do is you square this function
so take the absolute value square it and
what that's going to 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 going to find the
electron there points where there's a
large value of sy be it positive or
negative there's going to 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 going to 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
them here a lot of them there a lot of
them here and a lot of them here always
where there's these peaks you get more
of them then you would have at other
points where the values were smaller you
build up a distribution that's
represented by this wave function so the
wave function does not tell you where
the electron is going to be it just
gives you the probability and
technically the square of it gives you
the probability of finding the electron
somewhere so even it points down here
with 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 sold some Schrodinger equation or
we were just handed a wave function and
we were told it looks like this and
we're 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
here but you'd have the greatest chance
of finding it in this region right here
so you 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 is
going to 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 going to
do the experiment on in other words does
it describe one electron or only
described 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 a hundred
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 that for about a hundred years now
we've been able to make progress with
quantum mechanics even though we differ
on how exactly to interpret what this
wavefunction really represents so
recapping the wavefunction 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