 Hi, I'm Zor. Welcome to Inizor Education. Today we will try to solve a couple of very, very simple problems related to interaction between photons and matter. Well, as far as the matter is concerned, we will discuss only the simplest element, which is hydrogen. It's the simplest because it has only one electron. Okay, now this lecture is part of the course called Physics for Teens, presented in Unizor.com. There is another course which is prerequisite for this one, which is mass for teens. You definitely have to know mass before you study physics on the level which we are trying to present right now here. Now the site is completely free. There are no advertisements, no strings attached. You don't have to even sign in unless you want to do something which is called like supervised studying. Then you have to have a supervisor, etc. Then everybody has to have a sign-in. In any case, it's still free. Every lecture has notes on the website. Notes are basically like a textbook, which is part of a chapter of a textbook, which is dedicated exactly to what the lecture is about. So it's a lecture-video presentation, and next to it you have a textual part. What else is important? Well, it is important to watch this lecture as far as... You see, this is part of the course, which means it's better to go through the whole course. Lectures are kind of related to each other. If you have found accidentally this lecture through a search engine somewhere on YouTube, well, you definitely can watch whatever you want, but it makes more sense to watch it as part of the course because there is a lecture before that and material in that lecture will be used in this one. So it's all logically connected. There is a menu, a hierarchical menu. So the course is divided into parts. This part, by the way, belongs to... It's called waves. Then the part is divided into chapters. This is the chapter about photons and matter, and within the chapter you have a few lectures, including certain problems, etc. Also, in many chapters you will find exams, and you can take them as many times as you want until you get the perfect score. Okay, back to problems, simple problems. Okay, we did discuss before that electrons are circulating somewhere around the nucleus in shells. Now, the contemporary view related to quantum physics is that these shells can have only a fixed position relative to the nucleus, which means electrons which are circulating around these shells, within these shells around the nucleus, can have only fixed, discreet amounts of energy, potential energy. Well, whenever something is on a certain distance from something else and there is a force between them, there is potential energy, obviously, right? So if you have, for instance, a stone which you have lifted from the surface of the earth, we do some work, and that work actually is transferred into potential energy of the stone. If we will let it go, it will fall down, the potential energy will convert into kinetic and then into probably heat whenever it hits the surface of the earth. Same thing with protons and electrons. Protons are in a nucleus, electrons are circulating around it, and there are certain fixed orbits or fixed shells actually within which electrons can exist. Now, this is basically part of the quantum physics. I'm not going into kind of a why it happens. Well, many people don't know why. Most people don't know why. But that's the fact. And there are certain, you know, calculations, experiments, etc., which confirms this thing. And what's the most important thing is that whenever electron is jumping from one shell to another, its potential energy also jumps, either positively or negatively, depending on. So if we, well, electron is attracted to proton in a nucleus, right? Which means to move electron from a lower orbit to a higher energy orbit, we have to do some work, right? We have to separate them more, electron and proton, which means we have to do work. Now, if electron jumps down closer to the proton, it's not we who do the work. It's the electromagnetic field, the attraction between proton and electron are doing work. So that's why we are saying that the energy of electron growing when it goes to a higher level or diminishing whenever it goes to a lower level. When it's growing, it should absorb energy, absorb our work, whatever the work is. Most likely we are hitting the electron with photons, with light, with electromagnetic radiation. Absorbing this radiation, electron moves to a higher orbit and higher energy potential. Now energy potential, which it's moving higher, you see all the energy which electron has is basically a negative energy because we have to do no work bringing from infinity where potential energy is, by definition, is equal to zero close to the nucleus because there is an attraction. It's not we who should move it from infinity close to nucleus. It's electromagnetic field that does it. So that's why the energy is negative. If we have to do something, then the energy is basically growing. So whenever we are moving from lower orbit to higher orbit, the energy is growing while still being negative. And the further we move, the closer to zero potential energy is, right? Because potential energy on infinity is equal to zero, by definition, basically. Okay, so the energy is growing from negative with a higher absolute value to a negative to a lower absolute value. So the energy is growing. Now whenever electron moves back to the lower orbit, then the energy should actually be released, right? Because a certain amount of energy electron becomes more negative. So a certain amount of positive energy, like, for instance, electromagnetic radiation, or maybe we are hitting electrons so hard that it actually kicked out from the orbit and it has certain kinetic energy. So this is all transformation of energy. And what's important is that for this particular problem that energy is basically characterized by a certain discrete value on certain orbits. The higher it is, the greater the energy is, being negative. So when it's close to the nucleus energy, let's say minus 10, then energy is minus 9, minus 8, minus 7, et cetera, et cetera. Numbers are not whatever they are really. The real numbers, at least in case of hydrogen atom, were actually according to certain formula. Now this formula was suggested by, if I'm not mistaken, Nils Bohr, a famous physicist who basically made something which is called Bohr's model of the atom, which we will learn whenever we will talk about atoms. But meanwhile, what's most importantly is that the energy on the N's level, N's level being the shell number 1, number 2, number 3, et cetera, they are discrete. So we are numbering them. It's proportional to 1 over N squared. Now, in case we are talking about electron volts, as energy unit, which is kind of customary whenever we are talking about atomic dimensions, electron volts is, this is basically an exact formula for hydrogen atom. So this is hydrogen. Other elements do not have this formula, something much more complex or I don't know right now. So we will use this formula to find out, as the first problem, what's the differences in energy levels between different shells. And let's restrict ourselves to only four first shells. So here is my nucleus, and I will consider four shells around it. Three and four. So this is number four, this is number three, this is number two, and this is number one. So my first problem is, what's the difference in energy levels between these levels? Well, that's actually very simple. The N equal to one is a ground shell. So it's energy equal E1, N is equal to one, so it's 13.6 electron volts. Okay, now, what's the second? Well, the second is 13.6 divided by two square, divided by four. For the third, it's 13.6 divided by nine. And for the fourth, it's 13.6 divided by 16 electron volts. So these are energy levels. So what's the difference between energy levels? Well, I have a table basically. I, J, one, two, three, four, one, two, three, four. So what's the difference between energy levels? Well, between one and one is obviously zero, two and two zero, three and three zero, four and four zero. And then I just calculated how much is E1 minus E2? So that's the level of energy on the first shell, and this is the level on the energy of the second shell. And the difference happens to be, okay, it's 10.20, and this is minus 10.20. So it's from one to two, and this is from two to one. Whenever we jump down, energy is increasing. And whenever we go up, energy is consumed, that's why it's positive. Whenever we go down, the energy is released, and that's why it's negative, right? So two is more energetic level. Now, this is all minus, by the way. I forgot to put it. I've agreed that whatever electron has, it's always negative potential energy. So that's why such a difference. Now what's next? It is between three and one. So it's minus 12.09. So obviously from three to one, the difference in energy is greater than from two to one. So that's why we have it here, 13.009. What's next? The fourth is 12.71 minus 12.71, 75. And this is 12.75. So between one and two, as you see, the difference is significant. Between two and three is less. So what's between two and three? 1.89. Significantly less, minus 1.89. Then 2.55 was a minus sign. And this is 2.55. And significantly less will be between third and fourth. So the further we are, the closer to each other these shelves become. Well, obviously because the denominator is increasing and the difference between these two, obviously decreasing as well. And the difference is significant. So this is not proportional. If I wanted to do it proportionally, it would be like this. This would be the first level, this would be the second level, and this would be the third level. So you see the difference is closer and closer because the denominator is growing proportionally to n squared. So this is basically my first problem. I'm going to determine the differences between the energy levels of shelves. Why do I need it? Well, I need it because whenever electron jumps from lower shell to a higher shell, it's supposed to consume certain work. It happens when we are bombarding electrons with photons. And whenever electron is releasing its extra energy, so electron becomes excited whenever it goes from the ground level, from the number 1 to number 2, 3, 4, etc., it's excited. It has more energy than it usually supposed to have whenever it's in a normal state on the ground level. Then, electron, whenever it goes back to a normal, to a ground level, it releases the energy. And it releases exactly the same energy as there is a difference between the layers, between these shelves. Exactly one of these values. So if it jumps, for instance, from the third shell to the first one, it releases minus 12.09. It releases 12.09 electron volts of energy. Now, this energy is basically something like electromagnetic radiation, obviously. So it consumes electromagnetic radiation when it goes up the level, and it releases the same amount of energy whenever it goes down. Energy is supposed to be conserved, right? Okay, and we have agreed that electrons are jumping from one level to another whenever it consumes a certain amount of energy. Now, we were talking about the energy of electromagnetic radiation is delivered in chunks called quanta or photons. And the amount of energy according to Planck's formula is Planck's constant times frequency. So frequency of electromagnetic oscillations really is the key to minimum amount of energy when these particular oscillations, electromagnetic radiations can deliver. So they're exchanging energy between electromagnetic radiation and electrons in these chunks. So the amount of one chunk, which is this, of the light which bombards the electron one chunk is consumed by electron, it cannot consume less. It consumes in chunks, so it consumes one chunk and it jumps to the next level of energy. When amount of energy in this chunk equals exactly to the difference between these two orbits the electron is going from end to end. So whenever this amount is equal to one of these differences then the electron which has, let's say, it equals to 1.89 electron volts. One photon energy is equal to 1.89 electron volts. So whenever it happens and this photon hits the electron, electron jumps from the second to the third layer if there are electrons on the second layer, obviously. So it all is supposed to be in sync. The photons can hit and effectively move electron from one shell to another to a higher only when the energy exactly equals to one of these, which means that the frequency is supposed to be one of these. Now, what happens when everything is reversed? Whenever electron, because it's actually kind of bored to be on an excited level, it decided to go back to normal. Well, it releases exactly this amount of energy. If it jumps from the third to the second, it releases 1.89 electron volts of energy. Now, this electron volts of energy 1.89 should correspond to some frequency of light. So the frequency of light which can be obtained from this number, so if E is equal to 1.89 electron volts, H is a known constant. That determines the frequency of light which is emitted. And this can be visible or not visible, whatever the radiation can be. So that's very important. So my next problem is determine what exactly the frequency of light released when the electron jumps down from a higher energy level to a lower. How can I determine it? Well, first of all, to use this formula, you have to have everything in comparable units of measurements. So E is supposed to be J, H is known constant, and F is 1 over second, right? It's a frequency. So if E is J, H must be J time second. And then the F will be 1 over second, okay? That's what basically the balance of units supposed to be. So we know the value of Planck's constant. It's a very big number, and I have it in my notes for this lecture. But approximately, H is equal to 6.6, then 10 minus 34 J second. Now, electron volts. How can I convert electron volts into Js? Well, electron volts is amount of energy which is needed to move one electron from one level to another with a difference in energy, basically in the field intensity of one volt. Now, we know that J is one J is one cologne times one volt. Now, we don't have cologne, we have an electron, and charge of electron is supposed to be converted into colognes. And then by multiplying by one volt, we will get Js, right? So in colognes, my electron is E is equal to minus 1.6 times 10 to minus 19 cologne. So I have this. And now, I can calculate from electron volts, I will multiply electron volts by E, and that's my energy, whatever the energy is, expressed in colognes. Multiply by one volt, by one actually, and that gives me Js. So by multiplying this by this, this of this of this of this, I know the levels, or multiplying one of these numbers by this. I will have the difference between levels. And dividing by H, which I also know, I will get certain numbers which are the frequencies. So let me just give you the result of this. The results and frequencies are, so I have 2, 3, 4, 1, 2, 3. From 2 to 1, from 2 to 1, it's minus 10.20. So if I will multiply it by the charge of electron and divide by Planck's constant, I will have 2466. Now it's all multiplied by 10 to the 12. Now the 3 to 1 will be greater, obviously, because it's a greater amount of energy, and 30 H is free. Okay. Then, from 2 to 2, that doesn't matter. From 3 to 2, 457, and 617. From 4 to 2. And the 3rd. To 3rd level, I can jump only from the 4th, and that will be 160. You see how big the difference, by the way, is between frequency of a released light when it jumped from 4 to the first level, to the ground level, or from 4 to the 3rd, which is right very close to the 4th. It basically illustrates that the difference between layers, first from ground to the next one is a big one, then the smaller, smaller, smaller, and the difference between layers becomes smaller and smaller. That's because of this formula. It's n squared in the denominator. So these are frequencies, times 10 to the 12, by the way. Of the light, which is emitted when electron jumps from this layer to this layer. So from 2 to 1, it's this. From 3 to 2, it's this, etc. Fine. So these are the frequencies, times 10 to the 12. How to determine the wavelengths? Well, again, what is wavelengths? It's a distance which light, with a speed c, covers during the period. Well, period is the same thing as a reversed frequency, right? Time of one wave, and how many waves per second, right? So they are reversed to each other. So it's c divided by f. So this is my lambda. This is the wavelengths. Okay. So if I have the frequency of the waves, that's f, basically. How to calculate the wavelengths? I divide c, which is speed of light, which is approximately 3 to the 10 to the 8 meters. There is more exact number, by the way, which is again in my notes. It's 2, 9, 9, etc., etc. So I divide this speed of light into the frequency, and I will get the wavelengths. So what's the wavelengths? Here is the wavelengths. 122, 103, 97, 656, 486, and 1878. So frequency was the highest whenever we moved to the first layer. The higher the frequency, the shorter the wavelengths. So 122 is a very short wavelength. When we go from 3 to 1, it's even more energy we released, which means that the frequency is supposed to be greater, which means the wavelengths should be shorter, and the shortest one from the fourth level to the one. But again, to the second, which is much closer to the outside, the frequency would be significantly less, and the wavelengths would be significantly more. You see the difference here? And obviously the smallest difference between fourth and third layers, and that's why the frequency is the lowest, and the wavelengths is the largest. Now what are these wavelengths? And what's the unit? Now, unit is nanometers, which is 10 to the minus 9 of the meter. And the question is, do we see these lights or not? Well, there is a spectrum of visible lights. Apparently this one is red. This one is called cyan. I mean, there are many different spectrum of lights with names of the colors, which are different, because different people just see it differently. But approximately, this is the red and this is cyan. Basically, the shortest which we see is the violet. Violet is, let me put it here, violet. It's what? It's 380 to 440 nanometers. And that's the shortest wave which we see, nanometers. And the longest, which is red, from 625 to 740 nanometers. So in between, after violet, you have an increasing wavelength, blue, cyan, green, yellow, orange, and then red. But these are obviously very much subjective kind of opinions, so don't take it too seriously. Different people might call colors differently. From the physical standpoint, it's just a gradual change of the wavelengths. And again, in this particular case, for a hydrogen atom, from jumping, electron jumping from one shell to another, these are basically the wavelengths, which are, these are visible. These are ultraviolet, ultraviolet, which we don't really see them. They are much shorter. And this is infrared, which, again, it's like a heat, basically. We don't really see it, but we feel it with some other senses rather than eyes. Okay, so that's it. I do suggest you to read the notes, because there are some more precise numbers. And maybe I just put a little bit more explanation about how the energy is basically released when you're jumping, an electron is jumping from one layer to another. So these numbers are only for hydrogen atom. And this formula is only for hydrogen atom, the simplest one. More complicated, have a different story. And we might or might not, I'm not really sure myself, touch it when we will talk about atoms and their structure. But what's interesting is that there are distinct layers, distinct shells with distinct energy level for every element, not only for hydrogen. And this is all related to quantum physics. The whole energy view is quantized nowadays. And obviously, it's related to this formula that the frequency times Planck's constant gives you the amount of energy carried by one photon. So everything is divided. One more little detail. So these are kind of particle properties of the light versus wave properties. What's interesting is experiments actually show that particle properties of the light manifest themselves greater the shorter frequency is. I'm not right now going to basically go any further because that's kind of a complicated story. But you just have to understand that the longer waves, like radio waves, behave like waves more and less as a particle. The shortest, like ultraviolet for example, they really, we're talking about bombarding something which is a term obviously reserved for particle kind of thing. Corpuscular theory. Remember from Newton's time, he was the one who was basically explaining certain theories by a corpuscular model. Then we decided, you know, the physicists decided to go more to the waves and explained everything from the wave theory. And then came Planck and Einstein who started explanation of, let's say, Einstein explained the emission of electrons after it's bombarded by light, photo emission, photo electricity. He explained it from the particle standpoint, from the point of photons actually. Okay, that's it. Read the notes and there are no ex... Well, maybe there will be exams, I'm not really sure. Try to solve these problems yourself, by the way, and see if your numbers will correspond to the numbers in the notes. Notes have, to this lecture, they have correct numbers. Basically, try to come up with your own, basically, exercise in arithmetic. It's just arithmetic, but you know, have to do everything accurately and see if you have the same results. Thank you very much and good luck.