Deriving Planck's Law | The Equation That Began Quantum Physics

hello everyone my name is rishabh and this is the fifth video in the series of quantum mechanics in the last video we learned about the famous black body spectrum and the two experimental laws associated with it that is wayne's displacement law and stephen's law we saw how wean relay and jeans failed to explain the nature and the shape of the blackbody spectrum and then we discussed the ultraviolet catastrophe in today's video we are going to see how the nature and the shape of the black body spectrum was finally explained in the third attempt by a german physicist named max planck who is also known as the father of quantum mechanics we'll see how planck's explanation of the black body spectrum has a basic postulate of quantum mechanics embedded in it before we start make sure you subscribe to this channel and press the bell icon so that you don't miss any further videos of this series planck analyzed the problem deeply and also went through the work done by relay and genes really and jeans had assumed the black body radiation to be composed up of standing waves inside a cubicle box and they just calculated the number of modes of these standing waves or the number of modes of vibrations and came up with the following result number of modes to be 8 pi nu square upon c cube they assumed that these standing waves are created by oscillating particles which they call them harmonic oscillators now in order to get the desired result that is the energy density of the electromagnetic radiation or the blackbody radiation you first have to find the average energy of each of the harmonic oscillators from the laws of classical thermodynamics that number came out to be kt k is the boltzmann's constant and t is the temperature we discussed this in the last video also now in order to find the energy density all you have to do is multiply these two quantities so the energy density of the electromagnetic radiation in terms of frequency and temperature is the product of these two quantities 8 pi nu square kt upon c cube this is known as the rayleigh gene's law so rayleigh and genes came out with this formula of the energy density of the black body radiation but if you look at this formula you'll see it's a disaster why because when you integrate it over entire electromagnetic spectrum that is over entire frequency range this integral diverges so this means at higher frequencies say in the uv region and beyond the amount of energy emitted by a black body is infinite which is an absurd result also known as the ultraviolet or the uv catastrophe planck spent weeks to find a solution to this problem which he later described as the most strenuous work of his life planck finally found a flaw in this rayleigh jane's law there was no flaw up till here the number of modes of vibrations that rayleigh and jeans calculated was absolutely correct the mathematics had no flaw in it the problem was here in calculating the average energy of the harmonic oscillators rayleigh and genes used classical physics to calculate this result they had assumed that the harmonic oscillators which create the standing waves can have any amount of energy so the energy spectrum is continuous of the harmonic oscillators according to rayleigh and genes according to them any amount of energy can be interchanged between matter and radiation there is no restriction to it but planck said no he said that energy can only be exchanged in discrete amounts that is a standing wave cannot have any amount of energy a standing wave can only have energy in the multiples of h nu that is e n is equal to n h nu n is a natural number 1 2 3 and so on h is a constant known as planck's constant and nu is the frequency as usual now let us derive planck's law mathematically before we do that let us go back to the basics of probability first suppose i have a coin and i toss it what are the possible outcomes either i'll get a heads or i'll get a tails right how do you calculate the probability of a particular event we divide it by the number of total possible outcomes that is the probability that i'll get heads is 1 by 2 and so is the probability that i will get the tails so we have divided the desired outcome by the total number of outcomes now why do we divide this number by 2 that is why do we divide it by the total number of outcomes the answer is to ensure that the sum of all the probabilities is equal to 1. this is a normalization procedure in mathematics now suppose i want to calculate the probability that a particular mode of vibration has energy e n the formula for that is probability is equal to e raised to power minus e n by kt i'll explain from where this comes divided by summation n equal to 0 to infinity e raised to power minus e n by kt the exponential this is the probability that a particular mode of vibration has energy e n higher the energy of a standing wave lesser is the probability of its occurrence and that probability is proportional to this numerator e raised to power minus e n by kt now if you look carefully the numerator and the denominator are the same thing but the denominator contains a summation sign with it this is the normalization procedure that we just talked in the case of a coin so we have divided the particular outcome with the total number of possible outcomes this is the sum of the modes of all the modes of vibrations now the average energy carried by each mode is so average energy you take the energy e n multiply it with the probability and you take the sum over all the modes so after this it's just simple mathematics what i have to do is i have to just put the values of in this formula sigma remains same n equal to 0 to infinity e n now put the value of p n over here exponential minus e en by kt divided by this denominator i just substituted the value of p n e raised to power minus e n by k t now according to planck the value of e n is nh nu like we discussed just now so you put the value of e n e n is equal to n h nu over here sigma n equal to 0 to infinity e n is n h nu e raised to power minus again put the value of e n n h nu divided by k t e n is the energy carried by the nth mode of vibration so even in the denominator we substitute the value e raised to power minus n h nu by kt now in order to simplify the calculation we are going to assume e raised to power minus h nu by kt to be x so you put the value of e raised to power minus h nu by kt equal to x in this formula and you will get the average energy to be this summation remains the same n equal to 0 to infinity okay n h nu remains the same now this term changes to x raised to power n because out of all this term e raised to power minus h nu by k t is x so that becomes x raised to power this n is left so x raised to power n and similarly in the denominator sigma n equals 0 to infinity this becomes x x raised to power n so this is the result we have got after substituting the value of x e raised to power minus h nu by kt over here now i am going to expand this series first let me take out the constant since h mu is a constant it does not have any dependence on this summation and we can take it out of the uh summation so average energy simplifies to h nu and the rest remains the same n equals 0 to infinity n x raised to power n divided by sigma n equals 0 to infinity x raised to power n i have just taken out h nu out of the summation because it has no dependence on the summation now i'm going to expand the series one by one so what i'm going to do is i'm going to put the values of n 0 1 2 3 and then going i'm going to add the results so what we get is average energy is equal to h nu now put the values starting from n when i put n equal to 0 so this 0 x raised to power 0 i don't get anything i get 0 put n equal to 1 in the numerator 1 x raised to power 1 that becomes x plus because it's summation so we have to add each term now put n equal to 2 to x raised to power 2 this becomes 2 x squared and then n equal to 3 gives 3 x cube and so on similarly put the values of n in the denominator sigma uh now there will be no sigma so when you put n equal to 0 x raised to power 0 is 1 plus x raised to power 2 sorry 1 plus x plus x raised to power 2 this is x to the power 1 x cube and so on ok so this we have got by put by opening the summation now if you look carefully the numerator and the denominator they have become infinite series one last step to do is you take out x common from this numerator h nu x 1 plus 2 x i have taken x common out of the bracket 3 x square and so on in finite series and the denominator remains the same just simple mathematics nothing else now if you look at the numerator and denominator you will see that they are infinite series right they can be condensed into smaller versions that is the numerator i am going to write the series 1 plus 2 x plus 3 x squared plus so on this can be written as the binomial expansion of 1 divided by 1 minus x whole square so if you expand this term you will get this infinite series okay similarly the series the infinite series of the denominator 1 plus x plus x square plus x cube and so on can be written as the binomial expansion of 1 divided by 1 minus x so we have condensed these infinite series into smaller versions into smaller formulae right you put these values of the series over here what you get is average energy when when i substitute the value of numerator and denominator these two values over here i get average energy h nu x upon 1 minus x this is the value i get okay you divide numerator and denominator by x h nu because this x goes 1 by x minus 1 i have divided the numerator and the denominator by x and this comes out to be h nu x raised to power minus 1 minus 1 okay now put the back the value of x that is e raised to power minus h nu by kt in this formula you get this average energy h2 remains the same x raised to the power minus 1 x is this so this becomes e raised to power h nu by kt minus 1 this is the value of the average energy that planck calculated using his discrete energy spectrum now in order to get the final result that is the energy density of the blackbody radiation you have to just multiply this by the number of modes of vibrations that really and genes had calculated earlier so the final formula the planck's radiation law that is the energy density in terms of frequency and temperature is the product of this by the number of modes of vibrations so number of modes of vibrations were 8 pi nu square upon c cube multiplied by this value now h nu divided by e raised to the power h nu by kt minus 1 so this becomes finally 8 pi nu cube you multiply this h divided by c cube e raised to power h nu by k t minus 1 this is the planck's radiation lord is one of those equations in history that change the course of science this is the result that explains the black body spectrum the final formula the final mathematical function between the frequency and temperature that explains the shape of the blackbody radiation over the entire electromagnetic spectrum so this concludes the first part of this video in the next part of this video we'll see how we can shift back from quantum physics to classical physics and derive mathematically religion's law wins law and stephen's law from planck's radiation law see you in the next part

Channel: The Secrets of the Universe

Views: 17,618

Rating: 4.9603958 out of 5

Keywords: Planck's Law, Planck's Law Derivation, Quantum Mechanics, Quantum Physics, Science, How To Derive Planck's Law?, Rayleigh Jeans Law, Standing Waves, Black body, Black body radiation

Id: Yli7jT88_2Q

Length: 16min 19sec (979 seconds)

Published: Fri Oct 23 2020