 In this lecture, we will learn the following things. We'll learn about a classical model of the atom, synthesizing two semesters of introductory physics with some of the concepts that we've been exploring in this course. We'll learn about how to impose specifically the matter wave hypothesis on the atom and see if we can make some useful predictions about atoms using this structure, the so-called Bohr model of the atom. Let's briefly revisit the key observational features of the hydrogen atom at a macroscopic level. Atoms in general, when excited by an ionizing electric potential, emit not a continuous rainbow of colors, but rather a discrete set of colors, the so-called atomic emission spectrum. Shown here is the atomic emission spectrum for the hydrogen atom. It has a characteristically bright red line, which can be seen in the image over here on the right. It's got a blue-green or cyan line, which can also be clearly seen in the image over here on the right. It's got a darker blue line and a violet line, and those are a little bit harder to see. You can more easily see the dark blue and the violet line in this part of the image on the right. Now while the atomic emission spectrum can be readily revealed by ionizing a gas in a sealed tube, and then passing the light through a system that will spread it out, revealing the rainbow of colors that makes up any kind of light, the mystery of the atomic spectrum goes deep. Each atom has a characteristic spectrum. It's unique to each atom that we know of in nature. Hydrogen is different from helium. Helium is different from lithium. Each of them has this pattern that's their own. We do not understand the origin of this using classical notions of energy and momentum and matter, but at last we are ready to confront this last mystery left over from the 1800s using not only what was developed in the first two semesters of physics, but what we've been learning in this course. Now let's be more numerical about the hydrogen emission spectrum. We have a red line, a blue-green or cyan line, a dark blue line, and a violet line. These have associated wavelengths for the photons that carry each of these colors of light to our eye. The red line, for instance, has a wavelength of 656 nanometers. The cyan or blue-green line has a wavelength of 486 nanometers. The dark blue line has a wavelength of 434 nanometers. And the violet line has a wavelength of 410 nanometers. For something we'll do later in this lecture, it's worth noting these numbers down on a piece of paper. Go ahead and pause the video, write these four numbers down, noting the colors that go with each of them, and let's save that information for a little bit later. Now it was Johann Balmer, who worked out the mathematical relationship between these lines in 1885. These are the lines of light from the atomic emission spectrum of hydrogen that are visible to the unaided human eye. There are, of course, ultraviolet and infrared radiations from ionized hydrogen. We won't talk about those here, but they're represented in other spectra. The Balmer spectrum is the one that spans the range of light wavelengths that are visible to the human eye. Now Balmer noted that the wavelength of each of these lines is given by a simple formula, a constant b, times this ratio, an integer n squared, divided by that same integer n squared minus 2 squared, or 4. Plugging in n equals 3, 4, 5, 6, etc., Balmer was able to show that there is a clear mathematical relationship between these colored lines here. The constant b was determined to be 364.5 nanometers, and all one has to do to calculate the Balmer spectrum is know this number, and this formula, and use the integers greater than 2, and you can reproduce the wavelengths present in this picture. But why? Why is there a clear mathematical relationship between these colored lines emitted from hydrogen? Why are there similar mathematical relationships between the colored lines emitted from other atoms when ionized? These are deep questions, mysteries left over from the 1800s, that physics in its day could not explain. Now let's recall the earlier models of the atom that we explored in a previous lecture. Joseph John or J.J. Thompson, after discovering that cathode rays were really just electrons, and had masses that were far smaller than the latest known element at the time, hydrogen, constructed his Thompson model of the atom imagining that the electrons with their negative electric charges were embedded in a larger swath of positive charge spread out in space. And if one were to fire particles through such atoms, they would mostly miss the electrons which are very tiny and pass almost cleanly and undeflected through this diffuse positive electric charge. Ernest Rutherford and his colleagues, however, revealed by scattering alpha particles off of thin metal foils, that in fact sometimes the alpha particles would bounce almost directly back at the apparatus that had fired them at the metal in the first place. This implied that there was some kind of densely packed small core of positive charge at the heart of each atom, surrounded by orbiting electrons as if planets going around a sun. This model helped explain why, while many of the alpha particles would pass through the thin foil relatively undeflected, occasionally one of them would suffer a collision with this densely packed positively charged nucleus of the atom and suffer an immensely disruptive collision, some of which could send the alpha particles coming almost straight back at the source from which they had been emitted. Now all of this was happening in the very late 1800s with Thompson's work and the very early 1900s with Rutherford's work. And as you'll see, as we learned more about the atom, as people thought more deeply about the implication of Max Planck's adoption of the quantization of energy to explain the black body problem, and Albert Einstein's adoption of that same notion to explain the photoelectric effect, models of the atom changed rapidly in response to these ideas. And this then led to the ability to conduct calculations making new predictions about the behavior of atoms, but also new tests of how atoms should behave themselves. This was a dynamic period in physics, transitioning from the classical era of the previous three centuries into now the modern era that we would still be living in the after effects of today. Now let's consider the Rutherford atom, but let's simplify the calculations and only think about an electron going around a single proton, so a hydrogen like atom, but only in two dimensions. The electron is bound to the proton in a circular orbit by the Coulomb force in the same way that the earth would be bound to the sun by the gravitational force in our solar system. So the electron would be orbiting the proton, the proton would be the central force emitter in this problem, the electron would be responding to that force, the electric force, the Coulomb force in this case. So that force would be given here by this formula, the Coulomb force exerted on the electron by the proton would be given by a constant, one over four pi times epsilon naught, and I'll come back to that in a bit. Basically this is the permittivity of free space, epsilon naught, it has something to do with how electric fields can propagate through empty space. The product of the charges of the electron, which is negative the fundamental electric charge, and the proton, which is positive the fundamental electric charge, and that divided by the distance squared between the electron and the proton. Now this R would be the orbital radius of this circular orbit. Now I want to note here that this unit vector R hat carries all the directional information of this force. Now by convention R hat points from the source of the force, the proton, to the recipient of the force, the electron. It is the sign of the electrons negative electric charge that ultimately flips the direction of that vector and has the force pointing back toward the proton, that is making it an attractive force. Now according to Newton's second law, digging back to our first semester introductory physics, the sum of all forces on the electron will be simply summarized by its mass times its net acceleration. Well what acceleration is this electron experiencing as it orbits the proton? The answer is a centripetal acceleration. This ultimately is a center seeking force, which results in a center seeking acceleration, changing constantly the direction of the electron's velocity vector. So that means that the acceleration, the net acceleration of this electron has a well-defined form. It's given by v squared over R in magnitude, and its direction, center seeking, will point to the center of the circular motion, which again is in the direction of negative R hat, that is from the electron to the proton, whereas R hat is defined as being from the proton to the electron. So we have all the pieces we need to build a Rutherford model of the atom in two dimensions, using these ideas of a centrally compact positive charged nucleus and orbiting electrons. So let's go ahead and do that. We can set the sum of the forces, which is just the Coulomb force, equal to the mass times the centripetal acceleration. Now note that there is a negative R hat on the left side, negative R hat, negative R hat on the right side, negative R hat drops out of this entire equation. And we're left with just this equation, 1 over 4 pi epsilon not times e squared over R is equal to the mass of the electron times the velocity of the electron all squared. Now I'm leaving it in this funny form, because this almost instantly lets us write down the classical kinetic energy of this electron. That is 1 half m v squared. If I just multiply this equation by 1 half, I immediately get the kinetic energy of this electron going in orbit around the central proton. 1 half times 1 over 4 pi epsilon not times e squared over R. Now I'm going to leave this equation unsimplified. I'm going to leave this 1 half sitting out in front to ease the next step. And that is computing the total energy of this electron. You can go ahead and multiply this out if you want to. But it's convenient for what's going to happen next to just leave it out that way to remind us that doing math with this thing is going to be relatively straightforward. So let me rewrite the kinetic energy of the electron here at the top of the slide. Now the total energy is the sum of its potential and kinetic energies for that electron at a given orbital radius R. That is to say the total energy of the electron is just its kinetic energy plus its potential energy. Well the only force present is the Coulomb force. And so that means it has an electric potential energy. And that electric potential energy for the electron, Ue, will be given by its charge negative e times the electric potential of the proton, v with a subscript p. Well the electric potential of the proton is just going to be given by 1 over 4 pi epsilon not times the charge of the proton divided by the distance between them. So we wind up with this equation for the electric potential energy of the electron. 1 over 4 pi epsilon not times negative e squared over R. And that allows us to then write the total energy E as follows. It's just our kinetic energy plus the potential energy which is a negative number. And you see why I left the one half here. It was convenient because I have one half times a common multiplicative thing here and just subtracting off that common multiplicative thing here. And so ultimately at the end of the day I get a negative number for the total energy of the electron and that's okay. It just means that the potential energy of this particular electron outmatches its kinetic energy for this particular orbital radius R. So this is the final expression that I get for the total classical energy of a 2D Rutherford atom that is just considering the electron going around a stationary proton in the center. Let's write that equation down. We're going to pull it up later. We're going to need it again for something that happens later in this lecture but this is about as far as I'm going to go with this for the time being. Now how do we get modern concepts like the fact that the electron is actually a wave and not a point like particle into this thing? Well, we see the problem already with a Rutherford model. It's not going to explain the hydrogen emission spectrum, right? Because in the Rutherford model any orbital radius R is allowed. You can put any R in there and you'll get an E out. Now, because any total energy is allowed for the electron, this cannot explain the discrete energy spectrum of electrons in a hydrogen atom. We already have a sense that quantization of some sort must be present in the atom, right? This is the atomic emission spectrum. It bears the fingerprints of a constrained system with only specifically allowed energies determined by those constraints. Now, the de Broglie postulates enter to provide the crucial missing thing, the clue that will help us to understand this whole problem. This is the key step that ultimately leads to quantization. So for example, we know that the momentum of an electron matter wave is given by Planck's constant divided by the wavelength of that matter wave. We can relate the particle like properties, momentum, to the wave like properties, wavelength, using the de Broglie postulates. Now, classically, and because we're doing things at low velocity, we'll revisit this as a consequence of these choices later, but we're going to start off thinking classically. We can write the momentum in terms of its speed, and that's just going to be the mass of the electron times its velocity, or in this case the magnitude of its velocity at speed, thinking purely classically. So the question we want to answer is, is every wavelength of the electron possible for our electron? If it is, then every momentum is allowed in the atom, and we're right back where we started again. If every momentum is allowed, every speed is allowed, and if every speed is allowed, every radius is allowed. This has consequences in a system where there are relationships between things like speed and orbital radius, speed and momentum, and momentum and wavelength. But maybe that's the flaw. Maybe the problem here is that not every wavelength through a matter wave for our electron orbiting this central proton is actually allowed. We had a discussion in class about the Schrodinger of wave equation, and in that discussion I drew the real part of free particle wave functions or other kinds of wave functions up on the board, and I invited you, the class, to discuss whether or not those wave functions made physical sense. I mean, it's possible to write things down mathematically that don't make sense physically. And let's revisit that discussion. I will review the salient points here as I go through some examples. And let's apply that discussion to the electron in orbit around the proton and see what conclusions we might draw. Now, let's start by thinking about a circular orbit. A circular orbit is quite simply one that, after one period, repeats again. And that means if we're thinking about the electron as a wave at a moment in time, sort of frozen in space at a moment in time, remember it's not only going to be at one place, its wave function is spread out over space, and the space over which it's spread out is the circumference of its orbit. That's its one dimension that it's traveling along in this problem. So whatever the wave function of this electron is, it had better at least obey the basic mathematical principle that when it gets back to itself at the starting point of its orbital circumference, it starts all over again from exactly where it began. So let's imagine the real part of the electron matter wave, and we're just going to make one up. And it might describe the electron traveling along such a circumference of an orbit of radius r at a specific time zero. So we're going to freeze this wave function, which of course itself is not physical, but we're going to imagine it being frozen in time at a given moment. And it might look something like this. If we pick a zero point on the orbital circumference, which I've marked here in one dimension, it might be that its wave is at a local maximum. The real part of its wave function might be at a local maximum there. Then as we move along the circumference, the wave function declines and then goes negative, and then it comes back and it goes to zero, declining to zero again, and then it goes positive. Now, at one circumference, that is at 2 pi r, we see that the function I've chosen here nicely comes back to where it started. This seems to behave itself in the sense that the wave is one continuous wave that repeats nicely in space, because again, we've frozen in time. It's possible that this wave might be waving in time, but we've frozen in time, and so in space, this thing had better meet itself when it gets back to its starting point, and I'll explain physically why that needs to be in a moment. So this one seems to be a reasonable candidate wave function for describing our electron and orbit around a central proton. It nicely repeats itself when it reaches 2 pi r, that is the zero point again on its circumference. Its behavior is very smooth and continuous at the boundary where the orbit then repeats, where that is 2 pi which cycles back to zero again on a circle. Now let's take a look at a wave function that's also plausible, mathematically, but has some undesirable physical properties. So what about this matter wave for the same electron at the same radius r? We've frozen it in time. It's a perfectly reasonable wave function, right? It's got one wavelength here. It looks wavy. Is this a good physical wave function for describing the electron? Well, if we look at 2 pi r, we see that where the wave function ends up when it gets back to its beginning again is not where it started. Now that doesn't per se have any mathematical negative consequences. I mean this is a perfectly allowed function. I can write it down. It has something called a jump discontinuity when it gets back to its start. It jumps from this value right before 2 pi r back to its starting value at 2 pi r. But this has physical consequences. So because it has this jump discontinuity at 2 pi and does not cycle back to where it started, the jump discontinuity results in the first derivative at that point that is infinite. That is the derivative of this wave function with respect to space, d dx. At that point, 0 or 2 pi r has an infinite slope. Now because the first derivative with respect to space in the Schrodinger wave equation plays the role of the thing that tells you the momentum of the particle, the jump discontinuity means that we have an infinite momentum point in the wave function. And a place of infinite momentum is physically forbidden. It just doesn't make any physical sense. If this were the case, the universe would have ended long ago if things like this were possible, because there would be a particle which would contain more energy than every other particle in the universe, and that would have all kinds of terrible physical consequences. So this kind of wave function is physically forbidden. It may be mathematically allowed, but it violates physical notions of naturalness in the world around us because the jump discontinuity has a physical consequence that is infinite momentum. What about this wave function? What about this matter wave for the same electron again at the same radius r? Does this look to you like a good wave function? Go ahead and pause the video and stare at it for a moment. If you drew the conclusion that, yeah, it's a pretty good wave function, you're on the right track. I mean, it's got twice the number of wavelengths in 2 pi r that the first one did, but it comes back to where it started at 2 pi r. In fact, it differs exactly by a factor of 2 and wavelength from the first one. In fact, all waves that satisfy the relationship that their wavelength is an integer multiple of the shortest, continuous, and complete wave you can write down, the so-called fundamental, if you will. All harmonics of the fundamental wave of this electron will satisfy this condition that there's no infinite momentum point anywhere along the physical space, the circumference where it would occupy in space. And in fact, none in between those integer multiples will work. They'll all have the same problem that the previous example had. There'll be a jump discontinuity when you get to 2 pi r. This results in a place of infinite momentum. It's unphysical. So whatever the wave function that describes the electron in orbit around the proton, it must satisfy this condition in order to have physical meaning. An integer number n times some fundamental wavelength for lambda is going to be equal to 2 pi r. The only lambdas that will work will be those that satisfy this constraint, that is 2 pi r over n equals lambda. Now if we utilize the de Broglie postulate relating momentum and wavelength, then we wind up with n h over p substituting in for lambda equals 2 pi r. And classically remembering that p is equal to mv, this puts a constraint between the radius and the speed, and the integer multiple in question here. n h over mv equals 2 pi r. Now you'll notice that I can move the 2 pi over to the left side, and then I'll have h over 2 pi, and that allows us then to substitute with the reduced Planck's constant h bar. Remember that h bar is just h divided by 2 pi. You get this a lot when you start switching to the angular quantities, angular frequency, and wave number and things like that. The h bar is very convenient in those contexts. So let's go ahead and just absorb the 2 pi into the definition of h bar, and we'll arrive at the following equation for the speed of the electron and its relation to the radius of the orbit. That is m times v times r equals n h bar. That can be rewritten to solve for the speed of the electron, ve, which is n h bar over mr. And in preparation for relating this back to energy concepts, I'm going to go ahead and square ve, so I get ve squared, which is just this thing on the right hand side here. So from the matter wave hypothesis, I have a relationship between v and integer multiple of the fundamental wavelength and the radius of the orbit that determined that wavelength in the first place. Now, v and r also appear in things like kinetic energy. So you can already see that we have a new constraint to throw into energy equations that will lead us to perhaps some final understanding of why it is that the atomic spectrum is discretized. Now before we do that, I want to talk a little bit about Niels Bohr's actual postulate. It's worth noting that the way that Bohr attacked this problem was to postulate that there was a quantization of angular momentum in the atom. That is, the electron was quantized in its orbit around the proton. Now, he did this in 1913, and this was about 11 years prior to de Broglie's work, which was in 1924. So Bohr asserted, having, I guess, seen that quantization worked in other problems to explain things that had previously gone unexplained, he asserted that since H and H-bar, the reduced Planck's constant, have units of angular momentum, that is, joules times seconds, it might be in an atom that the angular momentum L is a multiple, an integer multiple of H-bar, that those would be the only kinds of angular momentum that would be allowed in an orbital system like a 2D Rutherford atom. So the angular momentum of the electron can only come in multiples of H-bar, and for a circular orbit we can relate the constraint and H-bar directly to the angular momentum of a particle going in a circle, and that's just P times R, which classically is MvR. And so this leads to the equation MvR equals NH-bar from Bohr's assertion. Now, later on, de Broglie would explain the reason why this works, and that's based on what we just saw in the previous slide. That is, if there's a constraint on the structure of the electron wave function, requiring that the radius, the circumference of the orbit be related to integer multiples of a fundamental wavelength of the electron, then if you go back to the previous slides, I'll go back in the lecture video, you'll see that this exact same condition resulted from the matter wave consideration. So this points to the fact that in 1913 when Bohr made this assertion, this is quite a bold assertion, really born out of the success of the ideas of quantization in the previous decade or so, based on Planck's work and Einstein's work and so forth. This was Bohr being very intellectually bold, and it paid off because, as you'll see, this model works extremely well. So finally we can take the kinetic energy equation and we can eliminate the speed of the electron in that equation in favor of the quantization condition from the matter wave hypothesis. So that is, here's our kinetic energy for the electron and orbit around a proton. That can be related to the generic kinetic energy, one-half mv squared. But we have an expression for v squared from the quantization condition from the matter wave hypothesis, and that was determined earlier to be this. So if we substitute that into the equation, we find out that the kinetic energy from Coulomb's law is equal to this kinetic energy expression taking into account the quantization of the wave function of the electron that only certain wave shapes will be allowed for a given orbital radius R. And some algebra will finally lead you to this expression for the allowed radii of an atom. It's actually quite remarkable. The allowed orbital radius in this 2D model is simply given by the product of an integer, 1, 2, 3, 4, et cetera, squared, that's n, times a product of a bunch of fundamental constants of nature. Notice there are no variables left. You have the numbers 4 and pi. You have the constant of nature, the permittivity of free space, epsilon naught, whose value is given here, 8.85 times 10 to the minus 12 Farad's per meter. You have the fundamental constant, Planck's constant, the reduced version, squared, and a reminder that h bar is 1.05 times 10 to the minus 34 joules seconds. You have the mass of the electron, 9.11 times 10 to the minus 31 kilograms, and the fundamental electric charge, 1.602 times 10 to the negative 19 coulombs. The only thing that can vary in here is n, and n is fixed to be an integer, 1, 2, 3, 4, 5, et cetera. So the radius of the orbit in our hydrogen-like atom is simply given by an integer squared times a number. So what is that number? Well, if we stick in n equals 1 and solve, we arrive at what is known as the Bohr radius. It's the smallest orbit allowed in the hydrogen atom because of the imposition of the matter wave hypothesis or Bohr's angular momentum quantization condition, which turned out to be equivalent. The Bohr radius is just this thing here, and if you calculate it out, it's about half an angstrom, 5.3 times 10 to the minus 11 meters, which is to say that the smallest hydrogen atom can ever be with its one electron and its one proton is about one angstrom across. And as we saw from earlier discussions in the lectures in this course, an angstrom is roughly the size scale of an atom, and that's no accident. It's imposed by the matter wave nature of the electron. So we find in this model of the hydrogen atom based on a classical definition of kinetic energy and momentum, but with matter wave quantization imposed, we suddenly find that only a fundamental orbit and its harmonics, or multiples of that orbit, are allowed. And this begins to look a lot more like the atom that gives rise to a quantized atomic spectrum. But the question is, can we see that spectrum arise from this model? Well, to answer this question, let's again consider the total classical energy of an electron orbiting a proton at radius r. But again, impose the condition that n times lambda, the matter wavelength of the electron, equals 2 pi r. That led to our writing of the Bohr radius and then all the allowed radii of our hydrogen atom, n squared times the Bohr radius. So we have the total classical energy of this electron. I'm just repeating the expression we wrote earlier. And then I'm plugging in with the expression for r, n squared a naught, and putting in the full definition of a naught here, the Bohr radius. So you wind up, if you play around with it a little bit, getting an equation that looks like this. You get negative, a bunch of numbers and constants, times 1 over n squared. That factor in front of 1 over n squared is negative 2.19 times 10 to the negative 18 joules. If you go ahead and punch in all the numbers here and calculate it. In electron volts, this is a much more familiar number. This is the famous negative 13.6 electron volts. This is the energy of the electron in the hydrogen atom in its lowest orbit. And of course, this also turns out to be the energy required to fully ionize an electron out of its parent hydrogen atom. If you want to free that electron completely, get it away from its proton, from hydrogen, and put it out at infinity, you have to put in 13.6 electron volts to liberate it. So the energy of an allowed orbit of integer n, corresponding to radius r equals n squared times the Bohr radius, is given by this simple equation. That the energy of that orbit, that specific orbit, is negative 13.6 ev times 1 over n squared, where n is 1, 2, 3, 4, etc., any integer. This is a remarkable fact. Just by imposing the matter wave hypothesis on this and requiring that the wave functions be well behaved when thinking about the wave function spread over the circumference of a circular orbit, we've immediately arrived at a quantization condition for our model of the hydrogen atom. But how good is this model? In order to understand how the Bohr model of the atom will give us the kinds of quantized energy spectrum that would result in specific wavelengths of light being emitted by an excited atom, let's step back and take a look at the Bohr model for a moment schematically. The picture on the left illustrates the classical drawing of what the Bohr model of the atom would look like. It's very similar to the picture that I sketched earlier with a single electron orbiting a single proton. Here, the smallest gray circle corresponds to the n equals 1 orbit, the smallest orbit that an electron can have around the proton at the center of the hydrogen atom. And that corresponds to the Bohr radius. The n equals 2 orbit is a multiple of 2 squared or 4 times the Bohr radius. And similarly, the n equals 3 orbit is going to be a multiple of 3 squared or 9 times the Bohr radius. Electrons can only orbit at these allowed radii, going in a circle around the central single proton. That means that if an electron is struck by, for instance, electromagnetic radiation, a photon, it could be caused to jump into a larger orbit if the electron possesses of sufficient energy to give the energy to the electron needed to transition from one orbit to the next. So, for instance, we might imagine that the electron started in the n equals 1 orbit of the hydrogen atom was struck by a photon of sufficient energy and was able to transition to the n equals 2 orbit of the atom. Maybe this resulted in a complete loss, a total absorption of the photon that struck it, or maybe the photon was scattered, losing energy and changing its wavelength, gaining a wavelength in the process, becoming longer in wavelength. Now, the image here shows the opposite of that process. An electron starts in, for instance, the n equals 3 orbit, and then spontaneously falls down to the n equals 2 orbit. But because conservation of energy has to hold, the energy difference between the n equals 3 orbit and the n equals 2 orbit must go someplace, and in this case it would result in the ignition of a photon. So, because the atom conserves energy, in order to go to a wider orbit, it must absorb the energy from someplace. A photon with the right frequency and wavelength can do that. To drop down to a lower orbit, that is one characterized by a smaller integer, n. It must release energy, and emitting a photon of a specific wavelength and frequency will do that, too. So, let's consider a transition that releases a photon, emits a photon in the process. From an orbit that's marked by an integer, n, greater than m, the orbit into which it falls. So, n is some integer, m is some integer, and n is greater than m. The change in energy, delta E, is going to be given by the final energy, the energy of the state marked by the number m, minus the energy of the initial state, the orbit marked by the integer n. Well, if we plug in the formulae for the energy of any specific orbit in a hydrogen atom, that's going to give us an overall multiplicative factor of negative 13.6 electron volts. And that's going to be multiplied by the difference between two fractions, 1 over m squared, minus 1 over n squared. So, for example, for the transition from the n equals 2 orbit to the n equals 1 orbit, or n equals 2 m equals 1, we find that delta E is E1 minus E2, and that's going to be given by 10.2 electron volts. Go ahead and work that out yourself for practice, but you should find that that energy is 10.2 electron volts. But this energy must go somewhere, and so this lost energy from the electron would go into the creation of a photon that then is emitted during the process, and that photon will have an energy given by h bar omega, the product of the reduced Planck's constant and its angular frequency. So let's think about the photon wavelengths from electron transitions in hydrogen. Using this energy conservation idea, and combined with the relationship between the frequency, wavelength, and energy of a photon, we can then calculate the expected wavelengths of photons emitted from an ionized Bohr atom. So an atom where, for instance, a photon starts out at infinity and comes down into one of the low orbits or maybe starts just above and drops down to a slightly lower orbit. Now recall that the Ballmer series, the visible wavelengths of light emitted in the atomic emission spectrum of hydrogen, involved an empirical relationship between wavelengths of emitted light from hydrogen given by the following formula, where the integer n ranges between 3, 4, 5, 6, and up. And this integer here is fixed at 2. Well, this looks a lot like the kind of relationship you might derive from the Bohr model of the atom in the transition between, say, n equals 3 and n equals 2, state. So just to see if we're at all matching reality, let's tabulate the energy of photons and the corresponding wavelengths of the photons that would result from transitions from the 3 to 2 state, the 4 to 2 state, the 5 to 2 state, and so forth. And if you do that, you find the following remarkable things. That the wavelength of the photon emitted when the electron goes from the n equals 3 orbit to the n equals 2 orbit is 656 nanometers. And if that sounds familiar, it should sound exactly like the red line in the Ballmer series, which has this wavelength. If the electron instead started in the n equals 4 orbit and dropped to the n equals 2 orbit, that results in a photon of wavelength 484 nanometers, which is blue-green, and is weirdly close to the blue-green line in the Ballmer series. Similarly, 5 to 2 results in a 432 nanometer wavelength photon that's blue, and 6 to 2 results in a 409 nanometer photon that's a violet. And these are, in fact, a good accuracy, the Ballmer series lines. Now they differ a little bit from the numbers before, and I'll comment on that in a moment. But overall, the pattern is very well explained by the quantization of orbits in the atom due to the matter wave nature of the electron, and thus the resulting quantization of angular momentum a la Niels Bohr's conjecture in 1913. This is a remarkable fact. The fact that just using a classical model of the atom combined with matter wave nature of the electron, one can immediately reproduce a pattern in the world around you. In this case, the Ballmer series of atomic emission spectrum lines. This is incredible. Now that said, it is wise to revisit our model and compare that to what we might actually expect from a more realistic model of atoms. After all, atoms are not two-dimensional things. They're three-dimensional things at the very minimum, and we haven't included an extra dimension in our model. We've only made a very good approximation to what we would expect real atoms to need to be more accurately described by, but you have to admit it's a pretty good model for what we were trying to accomplish. It almost exactly reproduces the Ballmer spectrum, which no previous model could do. So the so-called Bohr-Rutherford model of the atom, which is what we constructed here, has a few assumptions built into it. One of them is obvious. It's two-dimensional. We said that outright. A little bit less obvious, although I hinted at it throughout this discussion, is that we've modeled this atom as if the electron is free to move, but the proton, or substituting the proton with a whole nucleus with Z protons instead, so two, three, four, five, six, seven, eight protons, whatever you like, we have the proton pinned and unmoving at the center of the atom. But think about planets orbiting stars, or planets orbiting other planets, things of comparable size and mass orbiting each other. One isn't fixed while the other one goes around it. Rather, they co-orbit a common center, and that center is the center of mass of the system. So a more accurate model would take into account the fact that the proton can also wobble in response to being tugged on by the electron via the Coulomb force. Now, we've also obtained this model by combining a very classical picture of a planetary atom with very classical notions of momentum, kinetic energy, and so forth with the matter wave idea. That's how we stitched quantum physics into this through the matter wave idea. A more realistic model of the atom, of course, would be fully three-dimensional from the start. It would allow for the motion of both the electron and the proton, and in fact, if one puts that into this model, one much more accurately captures the bomb respect from wavelengths. They're a little off from what's predicted in this model, but they're almost exactly predicted by using a model where the proton can also wobble a little bit as it's orbited by the electron. And of course, in reality, we wouldn't start from a fully classical picture. We would try to exactly solve Schrodinger's wave equation in three dimensions, using, as our potential acting on the wave function, the Coulomb potential, written here in full three-dimensional glory. So there's our vector hidden in here as X and Y and Z in it. The truth is we are simply not ready at this stage to commit to more realism in describing the atom. This was already a bit of an exhaustive exercise at the level of, say, coming out of introductory physics, but I promise you that through the rest of this course we are going to build up a toolkit that would allow you to attack this problem in a later semester, starting from the principles outlined in this course. So let's review. We have learned the following things. We've learned about a way to develop a classical model of the atom from classical energy and force considerations. We've done so in two dimensions. We've then imposed quantum physics on this by sticking the matter wave hypothesis into the atom via the electron, thinking about what wavelengths would be allowed for an orbit of a given radius R, and then imposing that condition on the energy conservation that is derived from the classical model built in the first step. This has allowed us to make predictions about the behavior of a hydrogen-like atom in this model, and we found that it matches remarkably well with observational evidence. This is certainly a far more accurate description of nature than anything that has come before. And this is the Bohr model of the atom, which is a building block to a much larger picture of quantum physics, the physics of the smallest things in the universe.