I don't understand how Planck's equation can state that energy is quantized. in E = hv where E = energy of the photon h Planck's constant and v=frequency how can E have only discrete amounts? plancks constant is .. well.. a constant but isn't frequency a constantly varying or Infinitely variable quantity? If so, how can E have only certain values? (In the book "Advanced chemistry by Raymond Chang" it says that energy can come in packets which have energies of hv, 2hv, 3hv and so on... I'm having trouble understanding this as well..) thanks.
There are two innovative things: the energy quantization hypothesis (radiation and absorption by quanta) and the relationship between the quanta energy and frequency E = hv. The latter just implies that this relationship should be used in statistical treatment together with the the energy quantization hypothesis.
It means that electromagnetic radiation of frequency v can only come in "packets" of energy hv. Those packets are called photons. So what is quantized here, is the energy transmitted by electromagnetic radiation of frequency v. Of course, "electromagnetic radiation" can have any kind of energy, because, as you point out, frequencies are continuous. But for a *given frequency*, the energy exchange can only take place in jumps of hv.
In relation to OP's question: Is there a fundamental emergence of the quantization of electromagnetic energy? I can see why a "confined" electron has quantized energies (this just comes from the requirement that the electron wave must be fitted in the box) but unlike a photon, a free electron can have any energy ... Why does a "free" photon have to have quantized energies? (is there a theoretical justificiation for this?)
In order to obtain the quantum relationship E = hv, one can consider EMF amplitude for a certain harmonic. This amplitude obeys an oscillator equations. Quantum oscillator levels are quantized and the energy differences are just n*hv. Such oscillators are permanently coupled to charges. When an electron changes its level in an atom, the energy difference is given to/taken from a resonant oscillator => emission/absorption of a photon with E = E_{2} - E_{1} = hv_{12} = ћω_{12}. When many similar atoms make a transition together (in a laser), the resulting filed is a beam of many photons.
You have to be careful how you interpret the expression "photon energy is quantized". In fact, the spectrum of allowed energies of a free photon *is* continuous--i.e., a free photon *can* have any energy, whereas a confined photon's energy spectrum is discrete. This, however, is consistent with photon energy being quantized! As previous posters have noted, the literal meaning of "photon energy is quantized" is nothing more than the fact that a given photon's energy is directly proportional to its frequency, the constant of proportionality being Planck's constant: [tex]E = h \nu[/tex] But the question, "how is a photon's energy related to its frequency?" is a *different* question than the question, "what are the allowed energies of a photon?" The latter question depends on the physical situation; generally speaking, a "free" particle (meaning, approximately, that the particle is not "confined" to a limited space) has a continuous energy spectrum (meaning a continuous spectrum of allowed frequencies), whereas a "confined" particle (meaning, approximately, that it is not allowed to be anywhere in space, but only allowed to be in a certain compact region--this might also be expressed as the particle being in a "bound state") has a discrete energy spectrum. For example, electrons bound in atoms have a discrete energy spectrum ("energy levels"), whereas free electrons have a continuous energy spectrum, just like free photons. Historically, the "quantum hypothesis" was first put forward by Planck in order to solve the problem of the ultraviolet catastrophe in black-body radiation. Adding the quantum hypothesis, in that case, did *not* change the spectrum of allowed frequencies for black-body radiation--that remained continuous. What the quantum hypothesis did was change how hard it was to produce radiation of a given frequency; the old, classical assumption had been that all frequencies, even very, very high ones, were equally easy to radiate, whereas under the quantum hypothesis, low frequencies were far easier to radiate than high frequencies. So even in the original case, photons being quantized did *not* imply a discrete energy spectrum.
Just to make things clear: Suppose you have some electromagnetic pulse at frequency v, with a set amount of total energy E. The conjecture is that the energy is divided into quantized packets of energy: the photons. Each photon or energy packet carries an energy E_photon = hv. So total amount of energy packets or photons is simply E_total/E_photon. If we now increase the energy, then the only way to do this is to add another energy package. So we can have an energy of E_total and of E_total + hv, but nothing in between. This is the quantization what people refer to. We cannot have some electromagnetic field pulsating at a frequency v and containing some arbitrary energy E. The energy is quantized in terms of hv. You can punch a few holes in this story, but the main idea should be clear... ;)
I think you have provided the most careful answer, and it was helpful... So a free photon can have 'any' energy (because the spectrum variable f is essentially continuous), so the energy of an unconfined photon is in no way restricted... And the fact that electromagnetic radiation is carried by packets of energy, i.e, photon is analogous to the electric current being carried by electrons.
A free field has fixed periodicity T=1/v. Thus only frequencies v, 2v, 3v, ... nv are allowed and the energy come in packets hv, 2hv, 3hv, ... nhv, through the Planck constant.
So the basic idea is something like this. If there is a EM wave for a certain frequency (v), it will be made up of packets , each of which have a energy of hv. Am I right?
hmmm, this kind of remind me of quanta, an explanation for the ridiculous theory "an body emits energy at any frequency" which will result in emition of infinite energy... So a give atom or w/e will absorb and re-emitt the photons at a certain spectrum... Am i on the right track? I' am a bit comfused after reading this...
I still can't seem to understand this. Why go though all the trouble to say energy is quantized when in fact it doesn't appear to be so.. (at least to me.) From the above posts, here's what I understood. Photons CAN have any energy... However, their energy is proportional to their frequency. Therefore from a light source of given frequency(say, a red light) , there are many many photons rushing out.. all of them have the same energy because they have the same frequency.. But if slightly different shade of red was produced by the light, the light would still be made of photons.. however, the photons will have a slightly different energy. So isn't energy a continuous quantity?
Here is my understanding (most likely i'm wrong), in a laser radiation of specific frenquency each quantum carry's a certain amount of energy and there is no like.. in between.
Quantized means exactly what you've written in this case and yes energy is a continuous quantity since frequency is continuous. But that does not change the the fact that photons of a specific frequency all have the same energy which is a quantum mechanical result (in fact it was the FIRST result in what we now call QM), there is no classical "justification" for why this must is true, remember that there is no such thing as a "classical photon". Note also that this has important consequences not only for lasers etc but also for ordinary light, the fact that light is quantized changes its statistical properties (e.g. the distribution of light emitted by a source of a given temperature) quite profoundly which is why Planck was able to solve the UV paradox when he introduced quantization.
I would like to make a general comment: in many scientific disciplines, but especially in quantum mechanics, there are many "catch phrases" which are correctly and succinctly summarizing a certain property/phenomenon/event/view/... in a given context, but which are then thrown around in all generality and stated as "general truths" or "principles", especially in introductory and popularizing texts. It is my impression that they add more to confusing than anything else. "energy is quantized" is one of those catch phrases. There are many others floating around in "quantum speak". If you know what they mean exactly, then they are right, but usually one needs a much deeper understanding of quantum mechanics than is available to the reader to which one gives these "one-liners", and then they are genuinely confusing.