Lecture 6: De Broglie, Heisenberg, and Schrödinger

Related Resources

Slides (PDF)
Archived Notes (PDF)
Idle Mind Solutions (PDF)

» Download this transcript - PDF (English - US)

 

Let's get started. The weekend is over. Time to get back to work, back to learning. A few announcements. Tomorrow Quiz 2 based on Homework 2. Thursday will be the Periodic Table quiz. Remember, all you have to do is fill in the one or two-letter abbreviation for the chemical symbol.

And Friday is the last day of the contest, 5:00 PM. Send it to me by email. You can enter more than once, if you want. You can have multiple entries. You can do the lanthanides or the actinides or both.

Test 1 will be a week from Wednesday, and I will say something about that later in the week. And I will be available 3:30 to 4:30 today for office hours if someone wishes to drop in. Last day we talked about the validation of the Bohr model.

We saw that the Bohr model was able to correlate the observations of Angstrom, which had been formulated by Balmer. And, furthermore, the concept of energy level quantization was observed in mercury during the course of the Franck-Hertz experiment giving credence to the notion that quantization of energy levels is not the property of one-electron atoms alone but is something that is more broadly applicable.

So Bohr was validated in very, very strong measure, but there were also some contrary data. And we saw those. First there was the observation by Michelson who back in the late 1880s had done very precise interferal metric measurements of the hydrogen lines and had observed that the 656 nanometer line associated with the transition of n equals 3 to n equals 2 was, in fact, a doublet.

If you look really carefully, you could find it was not a single line but two lines, and Bohr is silent on such matters. And then we had Zeeman who conducted the experiment of hydrogen emission spectrum in a magnetic field.

And he found line splitting in the magnetic field. And, furthermore, that the intensity of the splitting was proportional to the intensity of the applied magnetic field. Again, the Bohr model silent about these splitting into multiple levels.

And subsequently Starck did the analogous experiment but with an electric field. Again, in an electric field line splitting observed, the intensity of the line splitting varying with the intensity of the applied electric field.

So there are some problems here. Sommerfeld came to the rescue in 1916. He preserved the broad concept of energy levels but put in some fine structure. And the fine structure involved taking those levels that Bohr quantified with the number n and assigning to those n levels a little bit of energy width so that there is a plurality of orbits, some circular, as in the Bohr case, some a little bit off circular.

And that bandwidth he called the shell. This is the shell model. And in order to label the various orbitals, as he called them, he introduced two more quantum numbers, l and m. And we saw l and m last day.

And then, at the very end of the lecture, we saw the last quantum number introduced, sort of a back to the future where we brought in the experiment by Stern and Gerlach where they sent the beam of silver atoms through an asymmetrical magnetic field and observed deflection, but deflection in two directions.

Deflection up and deflection down indicating that there was something strange that had yet to be accounted for. And several years later these two graduate students at Leiden by the name of Goudsmit and Uhlenbeck proposed that the electron in fact spins.

It doesn't simply orbit the nucleus, but as it orbits it spins. And since we don't know the absolute up and down in the universe, it is possible that some of the electrons may be spinning up. And, according to the right-hand rule, they would be spinning, by our convention, anticlockwise.

And some of them might be spinning clockwise, in which case, according to the right-hand rule we would consider them spin-down. And, in any arbitrary crucible full of molten silver, half of them are going to be spin-up and half of them are going to be spin-down.

And, sure enough, we see the splitting. And so that is where we left it on Friday. I thought, well, maybe we can now go to the Periodic Table and figure out what is going on there. If we look at the kinds of electron configurations we would expect, when we are at n equals one, which is k shell, we know that the selection rules allow for only one value of l, and that must be l equals zero, which the spectroscopists would call lowercase s.

And, under these circumstances, m must be uniquely zero. And s can take on values of plus or minus one-half. So there are really two choices here. So there are two electron configurations in the n equals one shell, if we follow according to the selection rules that we spelled out last day.

And, if we go to n equals two, this would be the l shell. And when n equals two, well, l could take on value of zero or one. And when it takes on value of zero then m must be zero uniquely. S can equal plus or minus one-half.

We have two electron configurations when l equals zero. When l equals one, m can vary over one, zero and minus one. S will always be allowed to take plus or minus one-half. So I have three times two is six.

All in all, in the l shell, I have the possibility of eight different configurations. And we will just do one more. And so when we get to n equals three that would be m shell by the spectroscopists' notation.

By the way, l equals one would be the p-orbitals. And so, in this case, l could equal zero, l could equal one or l could equal two. When l equals zero, again, m is zero and s is plus or minus one-half, so I can pick up two electrons here.

In other words, I have occupancy states for two electrons. The orbitals you consider as bins in which you can throw electrons, one per bin. Here I have m is one, zero, minus one. S again is plus or minus one-half.

There are six here. And when l equals two, m can vary from two down to minus two. S, again, is plus or minus one-half. So five times two is ten. This gives me, in total, 18. So this seems to be making sense.

And we talked about periodicity and properties, so let's go and take a look at what we have and see if we can reconcile this. If we look at n equals one, here is n equals one, hydrogen and helium. And, indeed, we have two electron configurations.

And then we are over to lithium. Well, lithium is n equals two. Here is 2s1, 2s2 and here are the six associated with the 2p. And then here is 3s, 3s2. And then something is going wrong. What I have here is there is the 3p, there are six of them, and now it is telling me I should be able to put in ten more.

But before I can put in those ten more, I am already over to 4s. Potassium is 4s, and I've only finished with 3p. Clearly, there is something else at work here. In other words, these energy states are not filling just in ascending n number.

We need to add something else in order to explain what is going on in the Periodic Table. And that something else is essentially the filling sequence. What is the filling sequence of electrons in orbitals? Just by n, l, m, s is inadequate.

There is something else at work here. It is close but not complete. And the answer to this question is given by the Aufbau Principle. The Aufbau Principle comes from a German word which means it is essentially construction or building, build-up principle.

And there are three components to the Aufbau Principle. The first one is the Pauli Exclusion Principle which was enunciated by Wolfgang Pauli. Pauli was an Austrian who did his PhD under Sommerfeld in Munich and then post-doced with Max Born in Gottingen and also with Niels Bohr in Copenhagen.

So you can see these people went to comparable schools. They worked with the same people in this group and kept the vigorous discussion going. He eventually got a position at the University of Hamburg where he was a Professor of Physics.

And what he said first of all is that in any system of electrons that the set of quantum numbers n, l, m and s, unique to each electron in a system. That is to say in a given atom. So each electron has a distinct set of quantum numbers, the first important idea.

You can think of each electron as having its own social security number or IP address or something like this. That is important. Once we have chosen a certain mix of n, l, m and s, it is used once for that particular atom.

So that is the first point. The second point of the Aufbau Principle is just energetics. Electrons will occupy orbitals in order of ascending energy, occupying the lowest energy first and up. Clearly, this n, l, m, s sequence is different from the energy sequence once we get beyond 3p.

So there is something else going on. Let's talk about electrons. Electrons fill orbitals from lowest energy to highest energy. But there is a wrinkle here. And that is that the energy levels themselves change with electron occupancy.

And as you go to higher and higher levels, those energy levels, as you saw in the case of hydrogen, are more closely spaced. The differences between those energy levels are becoming smaller and smaller as the n number rises.

And what the second point of Aufbau says is that as electrons begin to fill those levels the differences in energy may shift so that in an unoccupied state certain levels may be in the inverse order from how they are in the occupied state.

We have to recognize this. Lowest energy to highest energy and it is a function of occupancy. And we will see a few examples of that. That is the second part of the Aufbau Principle. And the third one talks about the question of degeneracy, and it is called the Hund's Rule.

The Hund's Rule talks about degeneracy. What do I mean by degeneracy? Degeneracy is where you have orbitals of equivalent energy. How do we put electrons into such a system? Orbitals of equivalent energy, we strive for unpaired electrons.

And what do I mean by that? Let's give a simple example. A simple example is, say, let's look at carbon. If we look at carbon, if you look on your Periodic Table you will see this notation next to carbon, 1s2, 2s2, 2p2.

What are we learning here? This first number here gives the value of n. So this tells me that this is n equals one. And s is this lowercase notation that the spectroscopists use in place of the l number.

This is s. S, according to spectroscopy, means that l equals zero. N equals one. L equals zero. And the superscript two is an indication of electron occupancy. In plain English, this says that there are two electrons in the 1s orbital.

And then we can continue. This is two electrons in the 2s orbital, and there are two electrons in the 2p orbital. Another way of displaying this is in a box notation. Here is 1s, if you like, k shell.

Here is 2s, l shell. And then there is 2p. And we know from over here that 2p has the possibility of three different m numbers. And I showed you last day that this is one case where trying to go in a Cartesian space makes sense, m equals one, zero and minus one indicates the three principle coordinate directions.

And, indeed, people refer to these as the 2px orbital, the 2py orbital and the 2pz orbital. So now let's fill the orbitals. Let's occupy the orbitals. And we are going to use the Pauli Exclusion Principle and we are going to use the fact that we fill from lower energy to higher energy.

So one goes spin-up and the other goes spin-down. Why spin-up and spin-down? Because the fourth quantum number is s. And Pauli says no two electrons in a given system can have the entire set of quantum numbers identical.

So this one is spin-up plus a half, this is spin-down minus a half. So it is compact notation. Likewise with 2s, spin-up, spin-down. I have two electrons now to occupy the p-orbitals. But these three p orbitals are degenerate.

That is to say they are of equivalent energy. So the Hund's Rule tells me how to put them in. And the Hund Rule says try to go for unpaired electrons. I will put one up here. And I cannot have an unpaired electron in the same orbital.

That would violate Pauli. So the only way I can get an unpaired is to put it alone in another orbital. And I don't care if you want to put it over here. I am not going to fight you over it. So the Hund's Rule is telling us to put the two electrons in separate orbitals unpaired as opposed to we should not put them in like this, like some librarian might want to fill nicely from left to right, up and down.

No. No librarian rule. Bad. Very bad. Somebody might be sort of a disorganized librarian and might want to put this one upside down but over here. Very bad. You violate the Hund Rule. If you violate the Hund Rule, you will have to talk about the appropriate physics punishment.

This is the Hund Rule and it explains electron occupancy. Now what we can do is look at an energy level diagram. And this is the energy level diagram for multi-electron atoms. And this is posted at the website.

Just to get you oriented, this is energy decreasing from zero to progressively negative numbers. Stability is at the bottom. N equals one, l equals s, can only be an s-orbital. L can only equal zero.

So we have two electrons here. Here is two plus six is eight. And now here is two plus six. And what we see is that 4s lies below 3d, according to this set of rules. And you don't have to know this by memory.

I would never ask you to tell me what the various energies are from heart. Instead, I would ask you given that this is the electron occupancy what is the explanation for it? So then you could explain in terms of these levels.

You can see 4s, then 3d, then 4p and then 5s. Let's see if that happens. Sure enough there is the 3s and then 3p. And now 4s jumps in before the 3d. Now these are the 3d elements. And now 4p. And now this is 5s.

5s lies below 4d. And, likewise, we get over to here and finally, look, here is 5d and 5f almost at the same level. And this is an example where unoccupied 5d lies below 4f. As soon as lanthanum takes that electron, this level rises ever so slightly, and then we jump down here and fill in the lanthanide.

But you notice up here how close these levels are, so we are looking at very, very fine distinctions. And here d, of course, can take the values from plus three down to minus three. That is seven different m values.

Seven times two is 14. Hence, you have the 14 lanthanide, 14 actinide elements. In other words, everything makes sense here. And now we can rationalize what is going on in the Periodic Table. Not on the basis of simple filling in ascending order, but a modified rate of filling as according to the Aufbau Principle.

Things are looking pretty good. Now what I want to do is take you exactly to this place where we are right now, where we see energy levels and electron filling, but I want to start all over again and I want to use a totally different approach.

I want to get to this same place now by wave mechanics. I want to get there by wave mechanics to arrive at the same place. And what is the same place? The same place is that energy is a function of these four quantum numbers.

And for that I need to invoke the names of three giants. And we are going to look at each of them and their contributions in turn. De Broglie, Heisenberg and Schrodinger. Let's go one by one. We are going to start with de Broglie.

De Broglie was the child of French aristocracy. He was thinking about doing a PhD in political science and pursuing a diplomatic career, but he was also drawn to science. And finally he decided to pursue a PhD in physics.

This was in Paris in the early ë20s. And he was very literary. The thing about his thesis was it a very short thesis. And what makes it great is that he began with an interesting question. You know the old adage.

If you ask a dumb question there is only a dumb answer. And the thesis is your statement. It is what you profess. Well, the thesis is also, another way to think of it, an answer to a question. So de Broglie's genius was in asking a really, really good question.

And then he answered it elegantly in a very small number of pages. I want to read sort of my encapsulation of his question. If a photon, which has no mass, can behave as a particle -- And we know this.

Sometimes we model light as beams of light, as rays of light. And sometimes we model light as a wave. And that doesn't bother us at all. Light can behave as a particle. In some cases it helps us to think of it that way.

Other times it behaves as a wave, and that is the way it helps us to think and rationalize what we observe. If a photon which has no mass can behave as a particle, does it follow then an electron which has mass can behave as a wave? 1920.

Bohr started talking about quantization. Now de Broglie is saying I am going to go further. I am not only going to say that we have quantization in the motion. We are going to say that the electron behaves as a wave.

One more time. It is beautiful. This is tautology. Ask not what you can do for your country, etc. If a proton which has no mass can behave as a particle, does it follow that an electron which has mass can behave as a wave? And, if so, he says this is what that electron wavelength would be.

Let's first of all associate him with this. De Broglie, 1924, in his PhD thesis says if an electron has wavelike properties this would be its wavelength. There is no hc over lambda. The electron is not moving at the speed of light.

He has to have some other expression for it. It is the ratio of the Planck constant to its momentum **lambda = h/p**. And this is just a Newtonian expression of momentum, the product of the mass of the electron times its instant velocity.

That is what he said. What are the consequences of that? How can we go somewhere with that? Now, you recall in Bohr the quantum condition. Bohr expressed the quantum condition by the angular momentum, quantum condition in the following manner.

Bohr said that the angular momentum, mvr is quantized where n is this integer counter h over 2 pi. This is the proportionality that is multiplied by the quantum. If de Broglie is correct, we could then model the electron in its orbit not moving as a particle, but let's model it as a wave.

So what kind of a wave is it? There are only two kinds of waves. There are traveling waves and there are standing waves. Well, this is in a stationary orbit so we need to have a standing wave. Not to scale.

Let's imagine this is the electron in its orbit. It is in a standing wave configuration. There is a geometric constraint. In order to have an electron in a stationary orbit this implies standing wave.

And a standing wave means there is a geometric constraint which is what? I have to have a whole number of wavelengths to get around the circumference. So the circumference is 2 pi r. And if it is going to be a standing wave then this must be an integral number of wavelengths.

But de Broglie has told us that the wavelength is related to the instant velocity through this formula. So let's substitute in. That will give us 2 pi r goes as nh over mv. And I can cross-multiply here.

That will give me mvr equals nh over 2 pi **mvr = nh/(2*pi)**. The quantum condition of Bohr falls out of the three line derivation if you accept de Broglie's hypothesis that the electron in this set of circumstances can be modeled as a wave.

So he got his PhD thesis, and in 1929 he gets the Nobel prize. He is off to a flying start. Einstein read the thesis and loved it. He loved it. But what do we care what Einstein says? Einstein is only a theoretician and de Broglie is a theoretician, so one theoretician propping up another theoretician, this is a mutual admiration society.

We need what? We need evidence. We need experimental evidence. Let's go back to high school for a moment, just by way of background. Remember these water tanks where you could put a vibrator in a water tank.

And this is top view of a water tank. And if you had a motor here that was causing a paddle to move up and down you could cause waves to form at one end of the tank and move from left to right. And, furthermore, you can do little experiments.

For example, you could put a dam in the middle of the tank. And if the distance between the wall and the dam was large in comparison to the wavelength then what happens? If this distance is large in comparison to the wavelength, this dam simply casts a shadow.

It is like light casting a shadow. So this is simple. When d is large in comparison to lambda the obstacles cast shadows. And this is equivalent to modeling the wave fronts as particles. This is ray optics.

This is the equivalent to ray optics, isn't it? You don't have to know anything about wave-like behavior. You have a front coming here. You have an obstacle. Where it can get by it moves through.

But in the other case where the gap, this d spacing, when the d spacing was small in comparison to the wavelength the obstacles did not cast a shadow. The obstacles did not cast a shadow. You did not see a tiny, tiny little stream coming out of each of these orifices.

Instead you saw this. Recall. You weren't sick that day, right? You were there. You remember this. Now, this is diffraction. And there is no way to explain this phenomenon if you model the water as having particle-like properties.

You have to invoke wave-like properties. Only by using wave-like properties as an explanation can you describe diffraction. Invoke wave-like properties to explain. De Broglie has said electrons, under certain circumstances, can be modeled as behaving as though they are waves.

What is going to be the experiment? We need to come up with an experiment in which the electron is going to be forced to behave as a wave. We are going to go to the United States, 1927, Bell Labs.

This is fast. Look, de Broglie was 1924. The thesis is written in French over in Paris. Somebody is reading the literature. They are top of this. And they do a beautiful experiment. What do they do as their experiment? People have been working with x-rays for about 30 years by this time.

And there is a technique known as x-ray diffraction. And x-ray diffraction allows us to characterize solid matter. It is so important, in fact, we are going to study it in 3.091 in detail next month.

But for now let's simply say that we have a diffraction which we know occurs only by wave-like behavior and it involves the use of x-rays. Why x-rays? Well, the wavelength of x-rays is on the order of about an angstrom.

If there is diffraction and the wavelength is about an angstrom, clearly the feature that the x-rays are diffracting from must be, likewise, dimension. It has got to be small. What is really small that would fit this? Atomic spacing.

Interatomic spacing is also on the order of about an angstrom. What Davisson and Germer did was constructed an elegant experiment. They took a crystal, this is a single crystal of nickel that has regular planes of atoms, and those planes are spaced on the order of an angstrom or less apart, and they irradiated this with x-rays.

This is h nu. X-rays coming in. And they looked at the result. And the result is a diffraction pattern that produces a series of rings. You have a single beam coming in, and you end up with rings.

And these rings are indicative of the crystal structure of nickel and can be explained only by invoking wavelike properties of light. Here is the critical experiment. If de Broglie is correct, let's irradiate the nickel crystal with a beam of electrons where the wavelength of the beam of electrons is identical to the wavelength of the x-rays that were used and see what we get.

So we use the de Broglie formula. And now we repeat the experiment, only this time we come with an electron beam. And the wavelength of the e beam is equal to the wavelength of the x-rays that were used previously.

The moment of truth. Photographic plate. What do they see? Circles. Electron diffraction was born. Electron. This is the e beam. There is no way that you could irradiate a crystal of nickel with a single beam of x-rays and get that circular ring pattern if the electron beam were behaving as a particle beam only.

So this validates de Broglie and it also validates the whole concept of wave-particle duality. Wave particle duality had been applied to photons. People are very comfortable talking about individual photons, but now we can talk about waves of matter.

And so if you say de Broglie's name to many physicists, the tagline they associate with de Broglie is "matter waves" as distinct from light waves. We have waves of light. We have waves of mater. That is important.

That is number one. I have to get you over to where we were earlier this morning. Number 2, we are going to Werner Heisenberg. What did he do? Well, he studied with Pauli. He and Pauli were buddies in graduate school.

They worked for Sommerfeld. You know, Sommerfeld never won a Nobel prize. But we should count how many Nobel prize winners he educated in his group. There was something very, very important going on in the mentoring process there.

Heisenberg gets his PhD with Sommerfeld in Munich and then goes to Copenhagen to work with Bohr. And he was getting burnt out so he decided to take a vacation. He went up North from Denmark to a deserted island of the coast of Norway and stayed there for about three weeks.

He came back, and with him he brought the Uncertainty Principle. And what he said was that when you get down to atomic length scales there are limits, there are constraints on our ability to observe.

At atomic length scale, limits on our ability to observe. And this is important because it means that since we cannot observe, we cannot talk about individual electrons. We are going to have to stop doing that because we cannot find an individual electron.

Every observation we make involves the exchange of energy. When I am looking at that wall, its photons are hitting the wall and coming back to me. And so the photons hitting the wall actually put pressure on the wall, but the pressure is so low.

Who cares? But you might say why does that make any difference? Well, suppose I want to look at something like an electron in orbit here. We know that this dimension is roughly one angstrom unit, right? The Bohr radius, for hydrogen is 0.529 angstroms.

Round numbers, this is one angstrom. Well, if I want to look at this, I want to have very accurate capabilities in terms of the energy that I use, the light that I use. So I would choose something that has a wavelength of roughly an angstrom.

You wouldn't measure the dimension of a human hair with a yardstick, so I need something on this order. If I take lambda equals one angstrom, go through hc over lambda, you will discover that the energy of a photon with one angstrom as its wavelength is on the order of 12,400 electron volts.

Well, I am looking for this electron. What is the binding energy of the ground state electron in hydrogen? It is 13.6 electron volts. So I am going to go and I am going to look for something that has an energy of 13.6 electron volts holding it in place, and I am going to use a flashlight with 12,400 electron volts.

I think I am liable to disturb the very thing I am going to measure. You say, well, why are you doing something so foolish? Why don't you use something that has much lower energy? OK, let's choose something that has, say, a hundred times less than that.

This will be 13.6 over 100 electron volts. Well, if I go through that, the wavelength of such a photon is going to be so big that I am back to measuring the dimension of the human hair with the yardstick.

So now you see the dilemma. If I want to get down to very, very fine dimensions, I have to bring huge energy and disturb the very thing I am going to measure. So this is the Uncertainty Principle. It leads to uncertainty in the measure.

Suppose I asked you to time a 100 meter spring in the Olympics, but I give you a stopwatch that is measured in units of 15 seconds? Well, these people are blasting through it, nine seconds and change.

You have a whole bunch of runners and five of them have all run it in under 15 seconds. I guess the whole batch is running under 15 seconds. And all you have is either zero or 15 because that is your resolution.

What does that tell you? It tells you that you cannot resolve what is going on at that level. In fact, there is something else you cannot resolve. You cannot resolve causality. When the uncertainty gets that big, you don't know which came first.

Cause and effect below a certain level of definition is blurred. Cause and effect, that becomes uncertain, too. This is philosophical, as a lot of this physics should be. What does it mean? It means that we better get away from these deterministic models where we have a little electron here with its potential energy and its kinetic energy.

And it is moving around and we are sitting on that electron watching it go about its daily affairs. Those models are not going to work anymore. What we see as a result of Heisenberg is the shift from deterministic models.

In other words, we cannot talk about individual particles. Deterministic models are now giving way to probabilistic models. Now, instead of talking about individual particles, we talk about ensembles of particles.

I don't know where any one of them is, but I can tell you on average where any one of them is likely to be. But I don't know where any one of them is. This means there is no more chicken and egg. It is chickenality and eggness, understand? Everything is blurred.

And to just give you a sense of what this is, the analytical expression that Heisenberg gave in one of its forms is that the uncertainty in the momentum of a particle times the uncertainty in its position, this is the uncertainty.

Not the value of the position. It is the value plus or minus something. This is the plus or minus, so the product of the uncertainty and the momentum. And since we are not expecting the mass of the particle to change, what we really are saying is the uncertainty in its velocity times the uncertainty in its position is greater than the ratio of the Planck constant divided by 2 pi.

He put a quantitative limit on that. And so I went through the math on this and said suppose I wanted to be really sloppy and I wanted to say if the delta X, the uncertainty in position is on the order of one angstrom.

In other words, I just want to know where the electron is somewhere within the shell radius of the ground state of atomic hydrogen anywhere. I don't even care if you try to put it at the heart of the nucleus.

Anywhere from plus 0.5 to minus 0.5 angstroms. All around. If you plug in the numbers, you will end up with an uncertainty in the momentum as being on the order of 15%. A 100% uncertainty in the position gives rise to a 15% uncertainty in velocity.

Well, what does that tell you? It tells you that any attempt to try to specify, to localize the position of an electron in such a system is futile. You are not going to be able to do it. And that is even without relativity.

It gets even worse. It is impossible to specify accurately the position of the electron. And the second thing is that the observer is part of the observation through that interaction. The way he described is when you try to get down a quantum dimensions and you are standing there with your camera, just remember the sun is at your back and your shadow is always in the picture.

You disturb the thing that you try to measure. That is number two. Now number three. Let's go to Erwin Schrodinger. Erwin Schrodinger also an Austrian. He was the University of Z¸rich and was getting burned out, too.

It was Christmastime 1925. He said I had it, so he headed off to a villa in the mountains, Christmastime 1925. Came back and just after New Year's 1926 and gave us wave mechanics. That is what he did over his Christmas vacation, wave mechanics.

So Christmas 1925 to New Year's 1926, what did you do for your Christmas vacation? Oh, I got wave mechanics. That is what I did. Wave mechanics. What he did was took de Broglie motion of the electron as a wave and developed a wave equation for such matter waves.

And the thing about what Schrodinger did was all he imposed on a system was the electron behaves as a wave and is bound. That is to say it is confined to the atom. He did not invoke the quantum condition, but he gets to the quantum condition.

That is the brilliance of Schrödinger. I want to give you an analogy. I am going to show the equation, but I don't expect you to solve it. You will have to have a little more math, a little more physics to do that, but I am going to talk about something that you are familiar with.

And that is a violin string. Let's take the situation where we take a violin string and we pluck it. I am going to graph this. This is the y coordinate and this is the x coordinate. The violin string extends from x equals zero to x equals L, and we pluck the violin string.

What do we have here? This is a bound system. It is bound. And so there are only two kinds of waves that can exist on this violin string, standing or traveling. Well, if this thing is going to vibrate for any length of time it better be a standing wave.

And what do we have here? Well, we know one solution to this is the following. I am going to solve it graphically and I am going to solve it analytically. You know, there is another solution. It could do this.

The ends are fixed. That is all I am doing. But there is another solution. I could do this. I could call this n equals one, the fundamental frequency. And I could call these others n equals two, n equals three.

These are harmonics. You know them as harmonics or overtones. And the mathematics of that equation involved a double derivative in time of x plus some constant times x equals zero with some constraints on it.

**x'' + k^2 x = 0** And the solution to this equation is y equals A cosine kx plus B sine kx **y = A cos kx + B sin kx**. All I have done is said I have a fixed system here. What comes out of the math is that the value of k, there are multiple values, there are multiple solutions.

I just showed you that you can have a single wave, a double, a triple and so on. It turns out it is n pi over l. **k = n*pi / l** The modes of vibration in that string are quantized. There is a plurality of solutions that conform to the boundary conditions, which is that the system must be bound.

What did Schrodinger do? Schrödinger said I am going to take that same idea, and I am going to apply it to a larger scale system. And what he does is writes this equation here. This is the Schrödinger equation.

It is a wave equation. It is in complex notation and so on. And the solution to this equation looks like this where it is written in terms of a quantity called a wavefunction. C is the wavefunction.

And the wavefunction is clearly nonphysical, but what we do know is that we can take the product of the wavefunction times its complex conjugate. And this is proportional to the probability of finding an electron.

And this is a function of space. Finding an electron in a region of space. This actually gives rise to a set of orbitals. The second thing is, just as in the case of the violin string, the wave equation, as posed by Schrödinger, has a plurality of solutions.

And the plurality of solutions have discrete values associated with them. But when you solve the Schrödinger equation, you don't get just a set of solutions that are dependent upon one number. You get a set of solutions that are dependent upon -- These quantum states fall out of the solution to this equation.

No condition imposed from the beginning that there must be quantized states. Instead, we have a certain energy, we have certain geometric constraints, we have a certain energy constraint and these fall out.

We have multiple solutions. These are termed the eigenvalues. In math, when you see multiple value solutions, these are eigenvalues. And these wavefunctions are the eigenfunctions. Now we are back to where we started.

We are back over to E as a function of n, l, m and s, only we got there through this other torturous route. And here are the samples of the wavefunctions. And you can see, we will take this up next day and we will have a look at what the orbital shapes look like as a result of plotting this.

See you Wednesday.

Free Downloads

Video

• iTunes U (MP4 - 111MB)
• Internet Archive (MP4 - 207MB)

Free Streaming

• VideoLectures.net