0:00:00.180,0:00:01.870
- [Instructor] So when people first showed

0:00:01.870,0:00:04.320
that matter particles like electrons

0:00:04.320,0:00:07.060
can have wavelengths and[br]when DeBroglie showed

0:00:07.060,0:00:09.620
that the wavelength is Planck's constant

0:00:09.620,0:00:12.280
over the momentum, people were like cool,

0:00:12.280,0:00:13.480
it's pretty sweet.

0:00:13.480,0:00:15.920
But you know someone[br]was like wait a minute,

0:00:15.920,0:00:19.235
if this particle has wavelike properties

0:00:19.235,0:00:20.721
and it has a wavelength,

0:00:20.721,0:00:23.690
what exactly is waving?

0:00:23.690,0:00:26.350
What is this wave we're[br]even talking about?

0:00:26.350,0:00:28.080
Conceptually it's a little strange.

0:00:28.080,0:00:31.010
I mean a water wave, we know what that is.

0:00:31.010,0:00:34.430
It's a bunch of water that's[br]oscillating up and down.

0:00:34.430,0:00:37.190
A wave on a string, we know what that is.

0:00:37.190,0:00:39.160
This string itself is moving up and down

0:00:39.160,0:00:40.920
and it extends through space.

0:00:40.920,0:00:42.240
But it's hard to imagine,

0:00:42.240,0:00:45.720
how is this electron having a wavelength

0:00:45.720,0:00:48.130
and what is the actual wave itself?

0:00:48.130,0:00:50.860
So physicists were[br]grappling with this issue,

0:00:50.860,0:00:52.556
trying to conceptually understand

0:00:52.556,0:00:55.940
how to describe the wave of the electron.

0:00:55.940,0:00:57.270
They wanted to do two things.

0:00:57.270,0:00:59.510
They wanted a mathematical description

0:00:59.510,0:01:01.240
for the shape of that wave,

0:01:01.240,0:01:03.090
and that's called the wave function.

0:01:03.090,0:01:06.740
So this wave function gives[br]you a mathematical description

0:01:06.740,0:01:08.390
for what the shape of the wave is.

0:01:08.390,0:01:11.210
So different electron[br]systems are gonna have

0:01:11.210,0:01:12.880
different wave functions,

0:01:12.880,0:01:14.730
and this is psi,

0:01:14.730,0:01:17.170
it's the symbol for the wave function.

0:01:17.170,0:01:20.010
So this is psi, the psi symbol.

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It's a function of x.

0:01:21.620,0:01:24.010
So at different points in x,

0:01:24.010,0:01:27.150
it may have a large value,[br]it may have a small value.

0:01:27.150,0:01:29.690
This function would give[br]you the mathematical shape

0:01:29.690,0:01:30.523
of this wave.

0:01:30.523,0:01:32.900
So that was one of the things[br]they were trying to determine.

0:01:32.900,0:01:34.580
But they also wanted to interpret it.

0:01:34.580,0:01:37.580
Like what does this[br]wave function even mean?

0:01:37.580,0:01:38.920
So we've got two problems.

0:01:38.920,0:01:41.430
We want a mathematical[br]description of the wave

0:01:41.430,0:01:44.490
and we wanna interpret what[br]does this wave even mean.

0:01:44.490,0:01:48.250
Now the person that gave us[br]the mathematical description

0:01:48.250,0:01:52.040
of this wave function[br]was Erwin Schrodinger.

0:01:52.040,0:01:55.090
So Schrodinger is this guy right here.

0:01:55.090,0:01:56.290
Schrodinger's right here.

0:01:56.290,0:01:58.420
He wrote down Schrodinger's Equation,

0:01:58.420,0:02:01.420
and his name now is basically synonymous

0:02:01.420,0:02:04.110
with quantum mechanics[br]because this is arguably

0:02:04.110,0:02:06.920
the most important equation[br]in all of quantum mechanics.

0:02:06.920,0:02:09.340
There's a bunch of partial[br]derivatives in here

0:02:09.340,0:02:11.150
and Planck's constants,

0:02:11.150,0:02:12.930
but the important thing is that it's got

0:02:12.930,0:02:14.610
the wave function in here.

0:02:14.610,0:02:16.470
Now if you've never[br]seen partial derivatives

0:02:16.470,0:02:17.810
or calculus, it's okay.

0:02:17.810,0:02:21.290
All you need to know for our[br]purposes today in this video

0:02:21.290,0:02:24.510
is that this equation[br]is a way to crank out

0:02:24.510,0:02:26.460
the mathematical wave function.

0:02:26.460,0:02:30.580
What is this function that[br]gives us the shape of the wave

0:02:30.580,0:02:31.920
as a function of x?

0:02:31.920,0:02:34.610
And you could imagine[br]plotting this on some graph.

0:02:34.610,0:02:37.950
So once you solve for this[br]psi as a function of x,

0:02:37.950,0:02:40.020
you could plot what this looks like.

0:02:40.020,0:02:42.120
Maybe it looks something like this,

0:02:42.120,0:02:44.010
and who knows, it could[br]do all kinds of stuff.

0:02:44.010,0:02:45.070
Maybe it looks like that.

0:02:45.070,0:02:47.260
But Schrodinger's Equation is the way

0:02:47.260,0:02:49.300
you can get this wave function.

0:02:49.300,0:02:51.380
So Schrodinger gave us a way to get

0:02:51.380,0:02:53.110
the mathematical wave function,

0:02:53.110,0:02:55.010
but we also wanted to interpret it.

0:02:55.010,0:02:56.140
What does this even mean?

0:02:56.140,0:02:59.750
To say that this wave function[br]represents the electrons

0:02:59.750,0:03:00.850
is still strange.

0:03:00.850,0:03:02.440
What does that mean?

0:03:02.440,0:03:04.150
Schrodinger tried to[br]interpret it this way.

0:03:04.150,0:03:08.600
He said, okay maybe this electron[br]really is like smudged out

0:03:08.600,0:03:12.430
in space and its charge[br]is kinda distributed

0:03:12.430,0:03:13.710
in different places.

0:03:13.710,0:03:18.010
Schrodinger wanted to[br]interpret this wave function

0:03:18.010,0:03:20.070
as charge density,

0:03:20.070,0:03:22.540
and I mean it's kind of[br]a reasonable thing to do.

0:03:22.540,0:03:24.900
The way you get a water[br]wave is by having water

0:03:24.900,0:03:26.560
spread out through space.

0:03:26.560,0:03:29.340
So maybe the way you get an electron wave

0:03:29.340,0:03:31.640
is to have the charge of the electron

0:03:31.640,0:03:33.320
spread out through space.

0:03:33.320,0:03:35.360
But this description didn't work so well,

0:03:35.360,0:03:36.820
which is kinda strange.

0:03:36.820,0:03:39.010
Schrodinger invented this equation.

0:03:39.010,0:03:40.370
He came up with this equation,

0:03:40.370,0:03:42.140
but he couldn't even interpret

0:03:42.140,0:03:44.140
what he was describing correctly.

0:03:44.140,0:03:45.190
It took someone else.

0:03:45.190,0:03:48.000
It took a guy named Max Born to give us

0:03:48.000,0:03:51.530
the interpretation we go with[br]now for this wave function.

0:03:51.530,0:03:55.040
Max Born said no, don't interpret[br]it as the charge density.

0:03:55.040,0:03:57.540
What you should do is interpret this psi

0:03:57.540,0:04:00.150
is giving you a way to get the probability

0:04:00.150,0:04:03.680
of finding the electron[br]at a given point in space.

0:04:03.680,0:04:05.040
So Max Born said this,

0:04:05.040,0:04:07.420
if you find your psi,[br]like he said go ahead

0:04:07.420,0:04:10.500
and use Schrodinger's[br]equation, use it, get psi.

0:04:10.500,0:04:14.980
Once you have psi, what you do[br]is you square this function.

0:04:14.980,0:04:17.010
So take the absolute value, square it,

0:04:17.010,0:04:19.820
and what that's gonna give[br]you is the probability

0:04:19.820,0:04:22.930
of finding the electron at a given point.

0:04:22.930,0:04:26.330
Now technically it's[br]the probability density,

0:04:26.330,0:04:27.590
but for our purposes,

0:04:27.590,0:04:29.400
you can pretty much just think about this

0:04:29.400,0:04:31.530
as the probability of finding the electron

0:04:31.530,0:04:32.400
at a given point.

0:04:32.400,0:04:34.870
So if this was our wave[br]function in other words,

0:04:34.870,0:04:38.070
Max Born would tell us that[br]points where it's zero,

0:04:38.070,0:04:40.710
these points right here[br]where the value is zero,

0:04:40.710,0:04:43.430
there is a zero percent[br]chance you're gonna find

0:04:43.430,0:04:44.400
the electron there.

0:04:44.400,0:04:47.820
Points where there's a large value of psi,

0:04:47.820,0:04:50.160
be it positive or negative,

0:04:50.160,0:04:52.500
there's gonna be a large probability

0:04:52.500,0:04:54.470
of finding the electron at that point.

0:04:54.470,0:04:57.210
And we could say the odds[br]of finding the electron

0:04:57.210,0:04:59.400
at a given point here are gonna be largest

0:04:59.400,0:05:01.630
for this value of x right here

0:05:01.630,0:05:04.536
because that's the point[br]for which the wave function

0:05:04.536,0:05:07.240
has the greatest magnitude.

0:05:07.240,0:05:10.450
But you won't necessarily[br]find the electron there.

0:05:10.450,0:05:13.400
If you repeat this[br]experiment over and over,

0:05:13.400,0:05:15.950
you may find the electron here once,

0:05:15.950,0:05:19.130
you may find it over here, you[br]may find it there next time.

0:05:19.130,0:05:20.800
You have to keep taking measurements,

0:05:20.800,0:05:22.140
and if you keep taking measurements,

0:05:22.140,0:05:23.630
you'll get this[br]distribution where you find

0:05:23.630,0:05:26.720
a lot of 'em here, a lot of 'em there,

0:05:26.720,0:05:28.590
a lot of 'em here, and a lot of 'em here,

0:05:28.590,0:05:30.867
always where there's these[br]peaks you get more of them

0:05:30.867,0:05:32.910
than you would have at other points

0:05:32.910,0:05:34.700
where the values are smaller.

0:05:34.700,0:05:37.530
You build up a distribution[br]that's represented

0:05:37.530,0:05:38.890
by this wave function.

0:05:38.890,0:05:40.610
So the wave function does not tell you

0:05:40.610,0:05:42.350
where the electron's gonna be.

0:05:42.350,0:05:44.640
It just gives you the probability,

0:05:44.640,0:05:46.800
and technically the square of it

0:05:46.800,0:05:49.510
gives you the probability of[br]finding the electron somewhere.

0:05:49.510,0:05:51.590
So even at points down here[br]where the wave function

0:05:51.590,0:05:53.020
has a negative value,

0:05:53.020,0:05:55.220
I mean you can't have[br]a negative probability.

0:05:55.220,0:05:56.950
You square that value.

0:05:56.950,0:05:59.770
That gives you the probability[br]of finding the electron

0:05:59.770,0:06:00.760
in that region.

0:06:00.760,0:06:01.593
So in other words,

0:06:01.593,0:06:02.680
let's get rid of all this.

0:06:02.680,0:06:06.160
Let's say we solved some[br]Schrodinger equation

0:06:06.160,0:06:08.230
or we were just handed a wave function

0:06:08.230,0:06:10.330
and we were told it looks[br]like this and we were asked,

0:06:10.330,0:06:12.710
where are you most likely[br]to find the electron?

0:06:12.710,0:06:15.450
Well the value of the[br]wave function is greatest

0:06:15.450,0:06:16.530
at this point here,

0:06:16.530,0:06:18.590
so you'd be most likely[br]to find the electron

0:06:18.590,0:06:20.200
in this region right here.

0:06:20.200,0:06:22.580
You'd have no shot of[br]finding it right there.

0:06:22.580,0:06:24.810
You'd have pretty good odds[br]of finding it right here

0:06:24.810,0:06:26.090
or right here,

0:06:26.090,0:06:28.110
but you'd have the greatest[br]chance of finding it

0:06:28.110,0:06:29.550
in this region right here.

0:06:29.550,0:06:31.703
So you'd have to repeat[br]this measurement many times.

0:06:31.703,0:06:33.440
In quantum mechanics,

0:06:33.440,0:06:35.890
one measurement doesn't[br]verify that you've got

0:06:35.890,0:06:37.220
the right wave function.

0:06:37.220,0:06:38.630
Because if I do one experiment

0:06:38.630,0:06:39.930
and measure one electron,

0:06:39.930,0:06:41.690
boop I might find the[br]electron right there.

0:06:41.690,0:06:43.230
That doesn't really tell me anything.

0:06:43.230,0:06:45.960
I have to repeat this[br]over and over to make sure

0:06:45.960,0:06:49.360
the relative frequency of[br]where I'm finding electrons

0:06:49.360,0:06:52.207
matches the wave function[br]I'm using to model

0:06:52.207,0:06:54.030
that electron system.

0:06:54.030,0:06:55.710
So that's what the wave function is.

0:06:55.710,0:06:57.360
That's what it can do for you,

0:06:57.360,0:06:58.650
although if I were you,

0:06:58.650,0:07:00.050
I'd still be unsatisfied.

0:07:00.050,0:07:02.590
I'd be like, wait a minute,[br]okay, that's fine and good.

0:07:02.590,0:07:05.117
Wave function can give us the probability

0:07:05.117,0:07:07.960
or the probability density[br]of finding the electron

0:07:07.960,0:07:09.050
in a given region,

0:07:09.050,0:07:10.640
but we haven't answered the question,

0:07:10.640,0:07:14.180
what is waving here and what exactly

0:07:14.180,0:07:15.510
is this wave function?

0:07:15.510,0:07:18.960
Is this a physical object[br]sort of like a water wave

0:07:18.960,0:07:21.070
or even an electromagnetic wave?

0:07:21.070,0:07:23.620
Or is this just some mathematical trickery

0:07:23.620,0:07:26.760
that we're using that has[br]no physical interpretation

0:07:26.760,0:07:29.776
other than giving us[br]information about where

0:07:29.776,0:07:31.590
the electron's gonna be?

0:07:31.590,0:07:34.310
And I've got good news and bad news.

0:07:34.310,0:07:37.500
The bad news is that[br]people still don't agree

0:07:37.500,0:07:40.160
on how to interpret this wave function.

0:07:40.160,0:07:42.520
Yes they know that the[br]square of it gives you

0:07:42.520,0:07:45.280
the probability of finding[br]the electron in some region,

0:07:45.280,0:07:48.110
but people differ on how[br]they're supposed to interpret it

0:07:48.110,0:07:49.200
past that point.

0:07:49.200,0:07:51.810
For instance, is this wave function

0:07:51.810,0:07:54.380
the wave function of a single electron

0:07:54.380,0:07:57.150
or is this wave function[br]really the wave function

0:07:57.150,0:08:00.750
of a system, an ensemble of electrons,

0:08:00.750,0:08:03.050
all similarly prepared[br]that you're gonna do

0:08:03.050,0:08:04.300
the experiment on?

0:08:04.300,0:08:06.730
In other words, does it[br]describe one electron

0:08:06.730,0:08:09.470
or only describe a system of electrons?

0:08:09.470,0:08:11.550
Does it not describe the electron at all

0:08:11.550,0:08:14.440
but only our measurement of the electron?

0:08:14.440,0:08:16.560
And what happens to this wave function

0:08:16.560,0:08:18.620
when you actually measure the electron?

0:08:18.620,0:08:21.120
When you measure the electron[br]you find it somewhere,

0:08:21.120,0:08:22.900
and at that moment there's no chance

0:08:22.900,0:08:24.480
of finding it over here at all.

0:08:24.480,0:08:26.689
So does the act of measuring the electron

0:08:26.689,0:08:30.720
cause some catastrophic[br]collapse in this wave function

0:08:30.720,0:08:33.280
that's not described by[br]Schrodinger's Equation?

0:08:33.280,0:08:36.240
These and many more[br]questions are still debated

0:08:36.240,0:08:37.980
and not completely understood.

0:08:37.980,0:08:39.289
That's the bad news.

0:08:39.289,0:08:41.940
The good news is that we don't really need

0:08:41.940,0:08:44.290
to understand that to make progress.

0:08:44.290,0:08:47.140
Everyone knows how to[br]use the wave function

0:08:47.140,0:08:49.490
to get the probabilities of measurements.

0:08:49.490,0:08:51.750
You can have your favorite interpretation,

0:08:51.750,0:08:55.100
but luckily pretty much[br]regardless of how you interpret

0:08:55.100,0:08:56.230
this wave function,

0:08:56.230,0:08:57.870
as long as you're using it correctly

0:08:57.870,0:09:00.070
to get the probabilities of measurements,

0:09:00.070,0:09:01.727
you can continue making progress,

0:09:01.727,0:09:03.540
testing different models,

0:09:03.540,0:09:05.470
and correlating data to the measurements

0:09:05.470,0:09:07.070
that people make in the lab.

0:09:07.070,0:09:09.040
Now I'm not saying that interpretations

0:09:09.040,0:09:11.500
of this wave function are not important.

0:09:11.500,0:09:15.060
People have tried cracking[br]this nut for over 100 years,

0:09:15.060,0:09:16.380
and it's resisted.

0:09:16.380,0:09:18.492
Maybe that's because it's a waste of time

0:09:18.492,0:09:21.500
or maybe it's because the[br]difficulty of figuring this out

0:09:21.500,0:09:24.270
is so great that whoever[br]does it will go down

0:09:24.270,0:09:27.230
in history as one of the[br]great physicists of all time.

0:09:27.230,0:09:29.130
It's hard to tell right now,

0:09:29.130,0:09:32.320
but what's undebatable is[br]for about 100 years now,

0:09:32.320,0:09:35.700
we've been able to make[br]progress with quantum mechanics

0:09:35.700,0:09:38.740
even though we differ on[br]how exactly to interpret

0:09:38.740,0:09:41.380
what this wave function really represents.

0:09:41.380,0:09:43.260
So recapping the wave function gives you

0:09:43.260,0:09:45.400
the probability of finding a particle

0:09:45.400,0:09:47.130
in that region of space,

0:09:47.130,0:09:50.090
specifically the square[br]of the wave function

0:09:50.090,0:09:52.610
gives you the probability density

0:09:52.610,0:09:55.510
of finding a particle[br]at that point in space.

0:09:55.510,0:09:57.660
This almost everyone has agreed upon.

0:09:57.660,0:10:00.600
Whether the wave function[br]has deeper implications

0:10:00.600,0:10:02.960
besides this, people differ,

0:10:02.960,0:10:05.190
but that hasn't yet[br]stopped us from applying

0:10:05.190,0:10:07.840
quantum mechanics correctly in a variety

0:10:07.840,0:10:09.403
of different scenarios.