00:01
So here we're going to say a few comments about particle wave duality as applied to quantum mechanics.
00:09
And then we'll take a look at some experimental evidence, as well as how you handle the waves theoretically.
00:19
But this is a modern idea that particles carry both particle properties and a wave associated with them.
00:27
What's a little bit abstract about the waves, well, not just a little bit, but very abstract, is that the waves only exist in a complex probability space.
00:39
So those wave functions give us the probability of finding a particle in any region of space, with a probability of its momentum being a certain value in that space.
00:56
So, first of all, momentum is inversely related to the wavelength of these probability waves.
01:06
And so you can create particles with a spread of momentum, and they will, for example, be deflected through slits.
01:17
They will interfere with not only other particles waves, but a particle by itself, its wave function will interfere if sent through a double slit situation.
01:36
So one of the more fascinating experiments is to send a single particle towards two slits and watch the evidence that these single particles have their waves interfere with themselves.
01:51
Okay, we won't get into that.
01:54
Theoretically, the wave function is handled with the schrodinger equation is one way to handle it.
02:00
Here we'll just start off.
02:02
Usually you start off with what's called the time -independent schrodinger wave equation.
02:08
And in that equation, conservation of energy is assumed.
02:13
Kinetic plus potential is equal to total energy.
02:17
With time -independent applications, the energy is a constant.
02:24
But the potential could depend on a variable x or r, for example.
02:32
The kinetic energy is going to come from the momentum squared over 2m, and the idea is that the momentum is inversely related to wavelength, and thus we're going to replace momentum with a momentum operator that looks kind of like a inverse distance.
02:57
Okay, so the momentum operator, again, has a h quantity in it.
03:04
H -bar is h over 2 pi, but then we are taking the derivative, something.
03:12
What are we taking the derivative of? we are taking the derivative of the wave function.
03:18
So it turns out the schrodinger equation, wave equation, like most wave equations, is going to turn out to be a differential equation.
03:28
Many of them turn out to be second order.
03:32
So we're going to put in the wave function.
03:37
You can be a function of x or it could be a constant.
03:45
And psi is known as the wave function.
03:52
And the product of psi times its complex conjugate gives us the probability density distribution.
04:04
Kind of like a gaussian would give you for experimental values.
04:11
So we'll write down the time -independent shredding or wave equation using the momentum operator squared just gives us a negative h -bar squared to m, then psi, then the potential energy, which could be dependent on a position variable, and all that equals e, a constant times psi.
04:40
So let's take a look at a couple of examples.
04:43
The first example is a fairly straightforward experimental application, but it's kind of a weird idea, is that you send a single electron into a double slit.
05:02
We'll make the slit separation d equals two micrometers.
05:14
And even if you send a single electron, a single electron in there at a time, what you'll see build up in your detector.
05:24
So you have a detector a certain distance away.
05:29
You will see places where the electrons hit.
05:34
So lots of hits.
05:39
And you'll see places where there's no hits.
05:41
So you will get this interference pattern.
05:44
And it works exactly the same way that light interference occurs.
05:52
So it's it's evidence that the electron carries a wavelength, and it's also evidence of the, sorry, evidence that the wave function exists in the sense that the single electron will interfere with itself and carries this pilot wave with it.
06:20
So the light equation is d sine theta equals m lambda, where m is the order.
06:29
Of the maximum.
06:32
So the first bright spot on either side is plus and minus 1, then plus and minus 2, with the middle one being m equals 0.
06:43
So let's figure out the wavelength lambda for m equals 2 bright spot with d equals 5 times 10 to the minus 6, sorry, 2 times 10 to minus 6 meters.
07:17
And theta 2 is 5 times 10 to the minus 6 radiance.
07:33
Okay, so it's not lambda for m equals 2.
07:35
It's the same lambda for all the electrons.
07:38
So they come out some momentum from the originating apparatus.
07:44
So lambda in this case is 2 times 10 to the minus 6 meters, and we'll use small angle approximation 5 times 10 to the minus 6 divided by 2.
08:04
And we get 5 times 10 to the minus 12 meters.
08:12
That is 5 picometers.
08:14
So very short wavelength there.
08:25
Okay.
08:25
Second application, we're going to be looking at using the schrodinger wave equation.
08:36
And we're going to, the easiest thing to do is assume some sort of wave function and see what system it works with, rather than starting with the potential and trying to figure out the wave functions that go with it, because you could get a fairly tricky differential equation to work with.
08:59
But here, if we were to look at the probability distribution for this, it would basically be a gaussian.
09:11
It would just be another exponential that, a square, x squared that peaks in the middle and has a certain width.
09:21
So the probability density is gaussian.
09:27
And we'd like to know what potential this goes with.
09:31
Does it correspond to a physical system? so we are going to be using the schrodinger wave equation, and, whoops, i see i forgot my second derivative in there, that's a bone -headed thing to do.
09:45
That doesn't look like a differential equation at all.
09:48
We need d squared, psi by d -x -squared.
09:54
Okay, so i'm glad i caught that...