Hello. I am trying to understand the problem of blackbody radiation but I don't have knowledge of advanced calculus (i know basic calculus) I know that experimental results do not agree with classical theory and I want to know more about the classical predictions. I also want to understand why emission energy would depend on frequency. I find this very counterintuitive. Does anyone know any good resources that are not "math heavy" but still give a an idea about the underlining concepts?
What you're referring to is known as the Ultraviolet catastrophe. I'll try to give an overview without using any maths or going too technical. In classical mechanics there's a principle which says the energy of a system is equally shared between all the different ways it can vibrate, known as 'modes'. These are essentially the different frequencies, so the first harmonic of a string vibrating has the same energy as the second harmonic and the third and the fourth etc. So how many modes are there? Well another principle says that the number of modes for a given frequency goes like the square of the frequency, so if you double the frequency then there's be 4 times as many modes at that frequency. So if the lowest frequency has some amount of energy, call it E, then at the doubled frequency there will 2 modes of energy E. So that is E+2E = 3E. Double the frequency and we get 4 modes of energy E. So now that is E+2E+4E = 7E. Then another 8E, 16E, 32E, on and on, up and up, summing ever more amounts of energy. There's no limit to the sum, it just keeps growing without bound and this is the UV catastrophe, warm cavities should emit infinitely large amounts of energy according to classical theory. The explanation for why this doesn't happen is that frequencies are not continuous, they are discrete and the new sum doesn't go 1+2+4+8+...., but rather the terms eventually stop growing and the sum of all the contributions is bounded. The precise value of the sum using Planck's work is what we observe from black body systems. I can be more detailed, more mathematical, if you want me to but you'll need to be specific about what you want to know and what your level of mathematical capabilities is.
I read the Wikipedia article on normal modes www.en.wikipedia.org/wiki/Normal_mode#Mode_numbers and I learned that it is possible for a system to vibrate at different harmonic frequencies and that a mode is a multiple of half a wavelength, did I understand this correctly? I also read the one on Equipartition theorem http://en.wikipedia.org/wiki/Equipartition_theorem, its idea somewhat makes sense to me; since the electrons of a blackbody at thermal equilibrium are vibrating and producing electromagnetic radiation then we could compare the energy distribution of the electrons to the Boltzmann distribution for molecules by saying that electrons will have the same average energy no matter what their frequency of vibration is and therefore all emission frequencies will have the same energy per wavelength (according to the equipartition theorem). Is this right? And from this resource http://hyperphysics.phy-astr.gsu.edu/hbase/mod6.html I understood that higher frequency waves have more ways to "fit" into the cavity so this is why there would be more modes the higher the frequency, and the energy for each frequency would depend on the number of modes. you said "frequencies are not continuous, they are discrete" did you mean energies? Please Register or Log in to view the hidden image! As to why energy quantization would resolve the problem I need more details. Thanks a lot for your input, it actually helped in making this completely new information to me make sense. Please Register or Log in to view the hidden image!
I'm also quite interested in the workings of Planck, it has never been clearly explained to me. For instance how Rayleigh-Jeans showed that the no. of modes for a blackbody cavity is proportional to the frequency squared, why more modes means more energy and the need for quantization.
I do not think that classical physics predicted that black body radiation would result in an infinite amount of energy as temperature increased. I thought it predicted that most of the energy would be at higher frequencies. Id est: Frequencies beyond ultra violet frequencies. Experimental evidence was known to refute this prediction. BTW: I think that Planck's paper had some errors or equations deliberately fudged to fit his view. Planck is usually given too much credit for founding Quantum Theory. Einstein's interpretation of Planck's paper & Einstein's paper on the photoelectric effect are the true foundations of Quantum Theory. I have been led to believe that Einsten was given the Nobelprize circa 1921 for the photoelectric effect paper as an apology for his being called a crackpot for his views on quantized energy. The idea is that all would know that his was the mind behind General Relativity, recognized as a tour de force & a revolution in the world of physics. Giving him the Nobel for his 1905 paper on the photoelectric effect was both an apology & an acknowledgement that he was the true originator of the notion of quantized energy or quantum theory. BTW: I think that some prestigious physicist of that era embarrassed himself in a letter of recommendation for Einstein who was applying for some position at a university. The letter asked that Einstein's weird notions about quantized energy be overlooked as a temporary mental aberration & that the focus should be on his other work. During the time he was viewed as a crackpot, a friend his opinion about the outcome of the controversy. He said something like the following.
