 Hi everyone. In this video I want to answer five questions. First, what is blackbody radiation? Second, what is the Rayleigh-Jeans law? Third, what is ultraviolet catastrophe? Fourth, what is the Planck's energy distribution? And fifth, what is the Planck postulate? Now, I have made videos on all these topics in a very elaborate fashion in the last three lectures but I have gone very much in detail individually in each of the videos involving a lot of derivations and mathematics. So, I am creating this summary video of blackbody radiation. In this video I want to discuss the summary of everything that I have discussed till now in my previous three lectures. I want to collect all the ideas in one concise video so that we can look at the progression of these ideas and take the necessary idea which is required for us forward in the future lectures. So, first of all, what is blackbody radiation? All the objects around us are emitting some kind of thermal radiation because of its temperature, because of the heat contained within them. Usually we do not see the objects glowing in our day-to-day life because the thermal radiation that they emit is usually lying in the infrared spectrum. But when we heat an object or increase the temperature of an object then usually the spectrum shifts towards visible light and then we see that the object has started glowing. This thermal radiation emitted by objects because of its temperature is known as blackbody radiation. Why is it called blackbody radiation? This is because in our day-to-day life the light coming from the surface of any object is a mixture of the thermal radiation and the reflected radiation. To filter out the reflected radiation we need to create a special surface that can absorb all the light falling on the surface so that the light emitted by the surface is purely thermal radiation. Therefore such an object would usually look black in color in room temperatures therefore the name blackbody and the spectrum is known as blackbody radiation spectrum. Scientists usually create a very specific kind of a structure to represent black body so they take a cavity some sort of a hollow metallic object and they have a hole on the surface so if a light falls through that hole then due to successive reflections in the inner walls of the cavity the light completely gets absorbed by that kind of an object therefore it behaves like a black body. Now if we heat this kind of an object to a certain kind of temperature the walls are going to start emitting radiation electromagnetic radiation and this radiation will be purely the result of its thermal temperature that the object has. This radiation is what needs to be studied and what experimentally gives us the blackbody radiation spectrum. So if we look at the experimental observations of blackbody radiation spectrum then the experimental observations are quite simple enough usually the graph is plotted with in the x-axis having frequency or in some textbooks they give you the wavelength so frequency and wavelength are inversely proportional to each other so you can use any one of the graph and in the y-axis you have the spectral energy density so this is the spectral energy density emitted by any kind of a blackbody density simply refers to per unit time per unit surface area or per unit volume so when we are doing experimental observations because the light is coming from the hole of the cavity so we take per unit surface area but when we are doing theoretical calculations because we are doing calculations within the cavity we take per unit volume but because the light coming from the cavity hole is ultimately a sampling of the cavity volume radiation therefore the spectrum is same for both the cases as far as the density of the spectral radiation is concerned. Now if we look at the variation of this kind of a spectral energy density with respect to frequency then for a small frequency the spectral energy density is very low it increases with increasing frequency reaches a maximum and again goes to zero it's a very simple looking graph there are certain kinds of experimental observations associated with it so for example if you calculate the area under this kind of a curve so you will end up getting the power emitted by a blackbody which is the energy per unit time it turns out is directly proportional to the fourth power of the temperature of the black body you see the black bodies are supposed to be in thermal equilibrium with the environment with some fixed temperature so whatever the temperature is the power is directly proportional to the fourth power of temperature so with increase in temperature the graph becomes larger and larger therefore the area under the graph keeps on increasing in this particular manner this is known as Stephen Boltzmann law there is another experimental law associated with this is that if you look at the frequency at which the power or the energy density is a maximum so if you look at this particular frequency if I call this as new max all right so new max is the frequency at which the energy density of the black body is at its peak or is at its maximum this new max is directly proportional to t that means with increase in temperature the peak at which the energy density you know reaches its maximum height shifts towards the right in a linear fashion and this is known as the Wayne's displacement law now I have done extensive discussion on this particular laws in my earlier video now I want to talk about the classical explanation of this kind of graph the first classical explanation or the most famous classical explanation for the black body radiation spectrum was provided by Rayleigh genes and Rayleigh genes called their expression the Rayleigh genes law I have done a derivation of the Rayleigh genes law in a very elaborate fashion in one of my previous lectures so what Rayleigh genes that was they considered a cubic metallic cavity and if this cubic metallic cavity is emitting electromagnetic oscillations within the walls then because these are metallic walls these electromagnetic oscillations end up forming standing wave patterns between two walls okay so these electromagnetic oscillations will have some kind of a condition because of this particular restriction that the length or the distance between any two parallel walls will be equal to some integer times half the wavelength so this condition is going to be satisfied because these oscillations are going to form standing wave patterns which comes as a consequence of classical electromagnetic theory so Rayleigh and genes used geometrical arguments to calculate the number of these standing wave patterns that were formed in some kind of a given frequency range per unit volume and they found that this came out to be an expression 8 pi nu square upon c cube d nu if you're interested in a derivation of this expression you can go back to my earlier lecture and check out that derivation essentially they came up with this particular expression for the number of standing waves for some given frequency range d nu and what they further did was they said that we can use this to multiply it with the average energy of the standing waves so they essentially borrowed something from classical statistical mechanics which is known as the law of equipartition of energy which says that for any kind of a system in thermal equilibrium so for in this situation we have the standing waves the average energy associated with the standing waves or the average energy associated with the harmonic oscillators on the walls of this cavity which is ultimately resulting in the standing waves comes out to be k t all right so if the average energy associated with the standing waves is k t and the number of standing waves for any given frequency interval is this much and therefore the total energy density emitted by the black body in terms of the energy of radiation per unit volume per unit time comes out to be the number of standing waves in a given frequency range multiplied by the average energy which essentially comes out to be 8 pi nu square upon c cube d nu k t if you look at this particular expression you see that this energy density is directly proportional to nu square so this is a parabolic relationship now what does the parabolic relationship tell us it simply tells us that for low values of frequency the Rayleigh genes expression might coincide with the experimental results but for higher values of frequency this expression is going to blow up because as nu tends to infinity this energy density for the black body cavity will also tend to infinity so essentially you end up getting some kind of a graph that looks like a parabola something like this so as you can see from this kind of a classical prediction it suits the experimental results for extremely low frequencies but immediately blows up at higher frequencies therefore creating a discrepancy between the classical predictions and the experimental results this brings us to what is known as the ultraviolet catastrophe it's called an ultraviolet catastrophe because it represents this kind of a discrepancy between classical physics and the fundamentals of classical physics and its failure to explain the experimental observations of black body radiation and this is something that happens at higher frequencies therefore it's known as ultraviolet catastrophe so this failure of classical physics to explain the experimental observations of black body radiations is known as ultraviolet catastrophe and therefore represents a very crucial point in the development of physics and its ideas because now we need something new or a modification of the earlier theories to explain why this discrepancy is happening in the first place so this is the Rayleigh genes prediction and these are the experimental observations all right so this brings us to the second last question which is the what is the Planck energy distribution so max Planck wanted to avoid this kind of an ultraviolet catastrophe and in his attempts to avoid the ultraviolet catastrophe he questioned the validity of the law of equipartition of energy which says that the average energy for the standing waves in the black body cavity or the average energy associated with the harmonic oscillators on the walls of the black body cavity is equal to kt in trying to fit the Rayleigh genes predictions with the experimental results he observed that the energy density was correctly representing the experimental results for low frequencies but it is blowing up for higher frequencies so he said that the average energy associated with the standing waves must therefore approach kt for low frequencies for new low but there must be some kind of a cutoff so that the average energy associated with the standing waves for large values of frequency must go to zero to avoid this kind of a blowing up of the function so he considered this particular condition that the average energy must be kt as predicted by the law of equipartition of energy for low frequencies but for high frequencies for some reason it should go to zero and he found out that this condition can be imposed by a very simple assumption about how the electromagnetic radiation is being emitted by the black body walls and that assumption is that the electromagnetic waves emitted by the black body walls is happening not in a continuous fashion but in discrete values of energy you see the electromagnetic waves emitted by the black body walls are essentially coming due to these oscillators oscillators means these electrons on the metallic walls they are oscillating or vibrating due to the temperature of that particular body and because they are oscillating and they are charged particles they emit that energy in the form of electromagnetic oscillations which ultimately is the black body radiation spectrum now in classical physics we assume that most oscillators or almost all oscillators in classical physics emit radiation or have energy in a continuous fashion so for example if you take a simple pendulum there is no discrete values of energy that the simple pendulum can have it instead has a continuous distribution of energy so a simple pendulum can oscillate with a certain amplitude greater amplitude means greater energy lesser amplitude means lesser energy so it can have a distribution of energy from zero up to large values in a continuous fashion similarly if you take a spring mass system a spring mass system can oscillate about its equilibrium point it can oscillate with low amplitude having low energy and greater amplitude having greater energy and this distribution of energy from zero to large values happens in a continuous manner and not in discrete manner therefore this is a very new idea because in classical physics most oscillators or most classical systems they have energies in a continuous manner but Max Planck suggested this radical idea that in this particular case of a black body cavity the electromagnetic waves that are being emitted by the walls are emitted in such a manner that the energy is coming out in allowed values of certain discrete numbers only so he suggested that the energy that is coming out by this kind of cavity walls is happening in values of let's suppose zero del epsilon twice del epsilon twice del epsilon and on and on so let's suppose this is the energy of the standing wave in a classical system it can go from zero to infinity zero to infinity in a very continuous fashion but Max Planck suggested that the energy emitted is instead emitted in some discrete numbers numbers of zero del epsilon twice del epsilon three del epsilon on and on del epsilon here represents the spacing between these energies so basically Max Planck assumed a uniform distribution of these kinds of energies so del epsilon represents the spacing between these allowed energies now what happens when you make this kind of an assumption if you make this kind of an assumption we can show that this particular condition comes out to be true how can we do that well this is because if we want to calculate the average energy of any kind of a classical system then what we usually do is we look at something like this so this is a graph where on the x-axis you have a continuous energy distribution and in the y-axis you have energy times probability associated with finding that system in that particular energy now what is the probability in classical physics we know of something known as the Boltzmann distribution the Boltzmann distribution basically tells us that if you have a system in thermal equilibrium then the probability that you will find a particle in a given energy is basically given by this kind of a probability distribution so for example if I take the case of molecules which are going around in a gas container then there are going to be a large number of molecules with low velocity there are going to be large number of molecules with higher velocity by velocity I mean kinetic energy so the probability of finding molecules with greater and greater kinetic energy decreases exponentially and the highest amount of probability is for molecules having lower energy or being motionless so this is a probability distribution which basically gives us an idea for any kind of a system in thermal equilibrium where vast majority of the particles may individually have different energies but the average itself is a constant so if you look at this kind of a probability distribution this also applies to a system like the standing waves in a blackbody cavity so if we want to calculate the average we look at the energy times the probability density versus the energy and you end up getting this kind of a variation and the average energy comes out to be a constant which is kt all right so for classical physics it comes out to be kt now what happens if instead of taking a continuous distribution of energies we only limit ourselves to some discrete values of energies here we can look at multiple cases where the spacing is small and the spacing is large if the spacing is very very small then you end up getting these rectangles associated with individual energies allowed energies right del e twice del e thrice del e etc etc and the average comes out to be somewhere near kt because for small spacing it it is similar to the entire graph the area of the rectangles basically gives us some kind of an average which is the same however if the spacing becomes large if del e becomes very very large then the average energy decreases keep in mind that if the spacing increases the average energy decreases in fact as the spacing becomes very very large the average energy tends to zero because the average is ultimately the area associated with all these rectangles corresponding to the allowed energies are you following what i'm saying for a continuous distribution the average is a constant kt for discrete distribution whether spacing between allowed energies is very low the average is again almost approximately equal to kt but for larger spacing you end up getting an average which tends to zero this crucial point is used by max plank to correct this particular expression because it satisfies this condition he said that for small values of del epsilon you end up getting the average is equal to kt and for large values of del epsilon you end up getting average tending to zero which satisfies his condition that he needed to fit the experimental data so now what he does is he tries to correlate del epsilon with frequency so for small values of frequency and small values of del epsilon you get this condition for large values of frequency and large values of del epsilon you get this condition so max plank assumes a relationship which we are going to call as the plank postulate where he says that del epsilon is a function of frequency and he associates a linear dependence of del epsilon on frequency so what he says is that del epsilon which is essentially the spacing between the uniformly distributed discrete values of energy according to which the radiation is emitted by the metallic walls this spacing is directly proportional to frequency or del epsilon is equal to some constant times frequency when he did that then the energy associated with the standing waves merely comes out to be 0 h nu twice h nu thrice h nu and on and on so energy an integral multiple of h nu where n can have values of 0 1 2 3 etc and using this in the calculation of the average energy so the average energy is ultimately the you know this expression ok where energy times probability associated with that particular energy some sum over divided by sum over all the probability associated with it when he made this calculation which I have by the way done in a very elaborate fashion in my last video you can take a look at that video if you are interested in this derivation but this expression comes out to be h nu upon e to the power h nu upon kt minus 1 this expression is very interesting because it satisfies this kind of an experimental result so let me show you how for example if we look at low frequencies for low frequencies what do we get here for low frequencies as nu tends to 0 e to the power h nu upon kt is equal to 1 plus h nu upon kt plus higher order term so you know the exponential relationship right so if you have something like e to the power x you can do a expansion of this kind of a function in terms of a series so that expansion is equal to 1 plus x plus x square upon 2 factorial plus x cube upon 3 factorial plus on and on so this is an expansion of the exponential function so if we make this expansion here but because frequency is very very less extremely small therefore the higher order terms will be extremely negligible therefore I can say that e to the power h nu kt is equal to 1 plus h nu upon kt which means that e to the power h nu upon kt minus 1 is approximately equal to h nu upon kt if I substitute this here then what do we get let me rub and show it to you if I substitute this here it simply ends up becoming this is the denominator so h nu upon h nu times kt comes out to be kt which is the result that the first condition here the first condition that means if I multiply this average energy with the number of standing waves I end up getting the classical prediction as well as something that is fits the experimental data so that means for low frequencies the planx postulate is sufficient enough to explain the results what about higher frequencies for higher frequencies we can also check so for higher frequencies is kind of obvious that as nu tends to infinity e to the power h nu upon kt also tends to infinity and therefore the denominator here tends to infinity and therefore because the denominator tends to infinity the average energy tends to 0 which is what is required for us to prevent this kind of a blowing up of a function so that means planx assumption about the discrete values of energy of the standing waves was successfully able to prevent this kind of an ultraviolet catastrophe from happening and if you in fact plot this function so if I take this expression and I put it here plug in here okay so this was originally the Rayleigh genes expression but now if I take the average out and I substitute this expression for average energy this comes out to be h nu upon e to the power h nu upon kt minus 1 this is the planx energy distribution that successfully satisfies the experimental observations of blackbody radiation because if you plot this it satisfies the radiation spectrum for low frequencies as well as for high frequencies thereby preventing the ultraviolet catastrophe issue the reason why plank was able to achieve avoiding the graph going to infinity at higher frequencies by assuming this was because I just told you using the Boltzmann distribution that higher energies are associated with lower probability because plank related the energy of the standing waves with frequency therefore higher frequencies were associated with higher energies and lower frequencies were associated with lower energies so higher frequencies although would contribute greater energy individually they would contribute less towards the average because their probability of occurrence is less and although lower energies would contribute lower energies individually those standing waves would together contribute more energy to the average because their probability of occurrence is more are you following so the low energy wavelengths or low energy frequencies had a greater probability compared to the high energy frequencies and as we saw from this particular graph the contribution of the higher energies towards the average is very less as epsilon average goes to zero for high frequencies but the contribution of the low frequencies towards kt is much much more and this brings us to the plank postulate which is what we want to carry forward in our next video the plank postulate is what max plank had to assume to prevent the ultraviolet catastrophe from happening which is that the energy emitted by the cavity walls in the form of standing waves or electromagnetic waves happened in discrete values of h nu where if i multiply by n it simply means that the energy is emitted as h nu 2 h nu 3 h nu 4 h nu etc etc this is a relationship of energy and frequency for the very first time we have a relationship of energy of electromagnetic radiation emitted by matter or absorbed by matter that is directly proportional to its frequency this is known as the plank postulate and this is also known as the concept of quantization because energy emitted or absorbed by the matter or the black body walls is happening not in a continuous fashion but in discrete values or in quantized values or in quantas so this is a condition for quantization quanta means discrete okay because the energies are discrete or quantized so this is known as a quantum so h nu is known as a quantum of energy so this is the condition for quantization and from here the word quanta comes all right based on which the entire subject is named we call it quantum mechanics because of this idea of quantum that means the energy associated with the radiation emitted by the surface or the absorbed by the surface is happening in discrete quantas okay so from here the word comes and defines the naming of the subject itself or the feel itself so this marks a revolutionary departure from the classical ideas of physics and we will take this idea forward into our next video where we will discuss the photoelectric effect where Albert Einstein would use this concept of quantization of energy of radiation in terms of explaining the photoelectric effect and we will see how that experiment succeed in explaining a different kind of phenomena so that is all for today I hope you have understood ultraviolet catastrophe the Rayleigh genes and the Planck's distribution and the concept of the Planck postulate so that is all for today thank you very much bye bye