having problems trying to understand and derive the equation for intensity in the double-slit interference pattern. any sort of help would be welcomed. thanks
Point Source I was asked by an old mechanical engineer how a photon that is emitted from a point source can then give up all its energy at a discrete point many thousands of diameters away. I gave an answer using the electromagnetic concept of radiation from a point source, thusly: Think of the electromagnetic wave as a sphere that keeps expanding and the energy distribution in the wavefront has to spread itself over a greater and greater spherical surface area. If you place an object somewhere in this expanding wavefront it will interrupt a portion of the expanding sphere (either absorbing or reflecting depending upon the characteristics of the object), the remaining wavefront keeps on going and going and expanding. The strength of an rf source dissipates with the square of the distance. Radiated energy can be starting at a point source and "spreading" three dimensionally. Take a volume of a sphere and double the radius, the volume is squared, thus the energy level at any point on the spheres surface is distributed over the spherical wavefront, and gets smaller and smaller at discrete points. ------ This really doesn't answer the basic question that was posed, and whether the energy of the photon is subject to the square of the distance rule.
Re: Point Source No, a photon is a unit that cannot be divided or spread out. Wherever it is absorbed, it is absorbed as a whole with its entire energy. However, an inverse square rule (like you described) can be used to understand how the probability of that photon striking a given point spreads out with distance.
Re: Point Source I’d like to express my opinion about this and then receive critiques from others. I might not be understanding this properly. It’s my understanding that a single photon or light “wave” goes out only in one direction and not as an “expanding sphere” of light. I think it could be described something like a long single (perhaps wiggling) bullet. However, when light bulbs radiate light and when radio transmitters radiate radio waves, a lot of these “bullets” are emitted out in all directions at the same time, and when all their different directions are plotted, they all go out in the form of a “sphere of radiation”. So, the sphere of radiation does not result from just one photon going out or a steady stream of them going out such as from a laser. If I am wrong about this, someone can please let me know.
Re: Re: Point Source Wouldn’t this idea match what I just said? Light from a light bulb dims at a distance because we are shining fewer photons per sq cm on a flat white wall at a distance, because the individual photons are spreading out the further away from the source they get?
Inverse square law In the following URL, equation (11) expresses the inverse square law for "intensity" and equation (16) expresses the inverse square law for "density". http://prancer.physics.louisville.edu/optics/manual/node7.html "Either a detector counting photons or one measuring energy will show an inverse square law with distance from the source." The following URL contains a "simple" explanation of the Inverse Square Law, http://www.astro.sunysb.edu/fwalter/CEN511/week2.html "All waves have the property that, once emitted, they propagate outwards from the source. As they do so the energy density in the wave decreases. This is because the total energy in the wave is fixed, while the volume of space the wave is expanding into is continuously increasing." Once a single photon is emitted from a point source in a particular direction, the photon has no connection to the source, and since it does not in itself expand, why would it lose energy? Once on its way it is self sufficient, correct? Equation (11) of the first reference implies photons lose energy. The value of P in equation (11) implies it refers to a group of photons, even though the source power emitted could be that of a single photon.
photon point source Equation (11) mentioned earlier is, [tex]F = \frac {P} {4 \pi r ^ 2}[/tex] where F is the detected energy and P is the emitted energy which is divided by what is the equation for the surface of a sphere. If I am concerned with only one photon from a point source and it is measured at a distance with a detector that is aimed at the incoming photon, there is no spreading out. Since it is not going to be measured with a detector that is spread out, the divisor is eliminated and the photon arrives at a distance with the same energy as emitted. P can be equated to the energy of one photon. Thus a single photon, whether it travels a thousand diameters or a thousand light years would arrive with the same energy that it started with. The basic question I was asked, is the energy of a single photon subject to the inverse square of the distance rule?
Re: photon point source Does the energy of a single photon get spread out according to an inverse square law? No. (I thought I had answered this, but perhaps I was unclear.)
Re: Inverse square law This was on one of your links: “Electro-magnetic waves expand spherically out into space.” I think this might be a common myth which has been confounding radio and electrodynamics students and old guys like me for many years. Surely a single photon doesn’t “expanding spherically out into space”. And even if a photon is also a kind of “wave” or a small long “wave packet”, a single one wouldn’t necessarily have to “expand spherically out into space”, not any more than a single writhing snake would have to “expand spherically out into space” if he were thrown in the air. If we throw out about 5,000 snakes in all directions: up, down, North, South, East, West, back, forward, left, and right, the whole group of snakes might “expand spherically out into space”, but all the individual snakes would not do that. Each one would go in only one direction, with that direction being altered only by gravity and air resistance.
Re: Re: Re: Re: Point Source Great! So, I wonder if we could say that sound waves become weaker at a distance because we are hearing or recording or receiving a smaller section of their spherical waves. Whereas light becomes weaker at a distance because we are receiving fewer side-by-side photons, since the photons are spreading apart as they move.
Re: Re: Re: Re: Point Source Doc Al, Since you are a smart guy, let me ask you something. This might be a very stupid question, but I’ve never understood mystery of the double slit phenomenon. In this demonstration, why wouldn’t the dot of laser light hit the card the slits are on, right in the middle between the slits, and thus not go through either slit? LINK TO JAVA DEMO
The slits are much smaller and closer together than in the demo you linked - microscopic even. Often, DIFFRACTION GRATINGS are used. The size/placement of the slits is on the order of the wavelength of the light.
Re: Re: Re: Re: Re: Point Source You are clearly a perceptive and intelligent person... As russ_watters points out, the demo is not to scale. That's not a stupid question at all: the laser must illuminate both slits or it won't work!
Re: Re: Re: Re: Re: Re: Point Source Doc Al and Russ, Ok, thanks. I have dozens of science books of all kinds that never mention that the slits are so close together a small laser dot of light can go through both slits at the same time, and every time I’ve had a chance to talk to a physicist or a physics professor, during the past 40 years, I’ve forgotten to ask them that question. Ok, now, the next question. What do they mean when they say a single electron is fired through both slits at the same time, or am I misunderstanding what they are saying about the electrons and the slits? Does the electron go through one slit or the other, or is it also split because the slits are so small and so close, and so it goes through both slits at the same time?
Re: Re: Re: Re: Re: Re: Re: Point Source Now you are asking the really tough questions that get to the heart of quantum mechanics! The same thinking applies to photons or electrons. In order to "understand" what's going on, you must consider the "wave" or quantum mechanical nature of the photon or electron. The wave function representing the particle must be such that it can be considered to go through both slits. I put the word "understand" in quotes, because if you mean how is it possible for a particle like an electron to go through both slits at once: as long as you think of a particle as a classical object (like a tiny marble) you cannot understand it! The great Feynman said: "The basic element of quantum theory is the double-slit experiment. It is a phenomenon which is impossible, absolutely impossible to explain in any classical way and which has in it the heart of quantum mechanics. In reality it contains the only mystery ... the basic peculiarities of all quantum mechanics."
Re: Re: Re: Re: Re: Re: Re: Re: Point Source Ok, and let me use a simple thought experiment to ask my next dumb question. My thought experiments are usually extremely simple. Let’s say we throw a watermelon through a picket fence. The watermelon is going to split apart and go through two or more of the slits between the pickets at the same time, yet a watermelon is not a “wave”. Could a particle-electron be splitting in the same manner, with half of it going through one slit and the other half going through the other slit? Suppose I think of an electron as a 19th Century watermelon that easily splits apart when thrown through a 19th Century picket fence? Apparently 21st Century watermelons might be considered to be “waves” rather than “solid” objects. Also, another question. Is the “mystery” of the electron going through both slits at the same time based on the idea that it is “impossible” for a “particle” to go through both slits at the same time, or is it based on the fact that an “interference pattern” shows up on the screen behind the two slits? Yet another question: Could the “interference pattern” actually be a pattern caused by the splitting apart of the particle-electron, with, in some cases, a bigger part going through the right slit, while the smaller part goes through the left slit, and in other cases, because the larger part goes through he left slit while the smaller part goes through the right slit, with the two parts being deflected slightly, in slightly different directions, each time a new electron is split while going through the two slits? I suppose this question leads to this question: Would a series of several watermelons thrown through a picket fence result in a debris splash pattern on the wall behind the fence that resembles an “interference-like pattern” that develops when a series of separate electrons are fired through two narrow slits? This may also sound like a dumb question, but I’ve seen such an “interference-like pattern” on walls and large cards, whenever I’ve spray painted things that have slits, that are located in front of the walls and large cards. Also, before I forget, I know a guy who had a peculiar problem regarding a narrow slit through which light passed in a special type of camera he made. He got an “interference-type pattern” with just one slit. We puzzled over this problem for months, trying to figure out all kinds of reasons for the cause of the pattern. Finally, I remembered the double-slit experiment, and I suggested that he might be getting some type of pattern because the slit was too narrow. So, he widened the slit and the pattern disappeared.
David, You asked me about the double slit experiment in another post, I think it is better to discuss the issue over here. At any rate, the double slit explanation given by quantum mechanics has not satisfied me, for many reasons. It appears to me, that quantum mechanics is really a statistical theory. Experiments performed using trillions of atoms in an atomic gas, lead to statements about the structure of a single atom. I don't think the reasoning there is correct. At any rate, I wouldn't mind really thinking hard about the double slit experiment if you want to. I will start out the discussion. Either no photons are point particles, or at least one photon is a point particle. Suppose that at least one photon is a point particle. We now have to deal with the possibility that all photons are point particles. So really then, we are looking for a model of a single photon. If we had a model of photons, as well as a model for the atomic structure of the diffraction grating, we might be able to explain the results of a double slit experiment. Now, let us set aside photon models, and think about the terms 'wavelength' and 'frequency'. The terminology is from the mathematical description of waves. And there are many different kinds of waves. Spherical, traveling, etc. Now, suppose that all photons are point particles. Then it follows that the wavelength of a photon and the frequency of a photon cannot be properties of photons. It would thus follow that they are parameters of the motion of a photon through space, and they would be related to orbital parameters of the photon before emission from the atom. Now trying to construct a model of photons, which models them as point particles is going against most of modern physics, but we can still analyze things logically. But here is my current point. Suppose that photons are point particles. It will then follow that if you fire a single photon at a diffraction grating, and the slit size is d, and the photon strikes an atom in the grating, that photon should be reflected. Now, if it is reflected with the same frequency, then its energy hasnt changed, and the collision was elastic. On the other hand, suppose that the photon entered the wall, for a ways, and then some random sequence of collisions occurred, resulting in many atoms emitting many photons, all of lower energy than the original photon. Perhaps the diffraction pattern might be explained by this "one causing many" idea. But in order to explain the double slit experiment this way, we would have to really understand what happens to the original photon at the atomic level. To properly explain the double slit experiment is going to take a whole lot of effort. I say we analyze things in stages. Stage 1: Are photons point particles? Yes or no?
Re: Re: Re: Re: Re: Re: Re: Re: Re: Point Source There are several key differences between the watermelon and the electron. When the watermelon splits apart it is essentially destroyed: all we detect are pieces of watermelon. But with electrons we never detect anything but whole electrons. Also, the watermelon does not display those pesky interference patterns. Yes and yes. For the reasons I gave above, I do not think this is a viable explanation. Not if by splitting you mean the physical splitting of an electron into two electron "pieces". No. The pieces would form a simple pattern (lots of debris where the slits are; little debris where the slits aren't) but it wouldn't be an interference pattern. In an interference pattern you'll get maxima right behind between the slits---where you would expect few particles to reach. I would expect you to see the same sort of pattern as with the watermelon. Excellent thinking on your part. You were most likely witnessing the infamous single slit diffraction (interference) pattern. Look it up!