Two Physics Questions

Factual Questions
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Question #1: Newtons Prism
We learn in high school physics that if you shine white light through a glass prism, colored (monochromatic) beams of light will emerge at different angles from the prism. We also learn that in order for a beam of light to have a well defined color (frequency) it must be very long. For a perfectly defined frequency a beam of light would have to be infinitely long.
Shine a short pulse of white light through a prism. In order for the above to be true the monochromatic beams of light would have to be emerging from the prism long before, and long after, the short pulse of white light goes through the prism. Can anyone clear this up?

Question #2: Diamond crystal
We learn in high chool chemistry that a diamond consists of carbon atoms each bonded to four neighboring atoms. The bonds must be at tetrahedral angles to each other so the four neighboring atoms cannot be all one one side of the carbon atom. This can only be true for an infinitely large diamond. What happens at the corners, edges and surfaces of the diamond? Are there bonds dangling out of the crystal, bonded to nothing?

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Question 1: For a perfectly defined frequency, you do indeed need an infinite beam. However, for a really well-defined, but not quite perfect, frequency, you need a really long beam. “Really long”, in this case, is defined in terms of the wavelength of the light. Visible light has a wavelength of a few hundred nanometers, and any prism you’ll encounter is much, much bigger than a few hundred nanometers, so you can do a very good (though not quite perfect) job of pinning down the frequencies coming out of the prism.

Question 2: They’ll be bonded to other things at the surface (at a guess, mostly hydrogen). But the number of carbon atoms right at the surface of the crystal is very, very small compared to the number of atoms in the interior, so it’s still a very good approximation to say that a diamond is pure carbon.

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1.) True monochromatic beams are an abstraction that, as you seem to suspect, don’t exist. As one professor noted – in order to have a truly monochromatic laser beam, his laser would need to be on forever (and to always have been on) with no fluctuations in intensity during eternity. “Not in MY lab!” he retorted.

The reality, of course, is that you can get quasi-monochromatic light from a source of finite duration. Heck, other factors involved in generating light (the widths of the energy levels, relative velocities of moving sources adding Doppler effects, thermal motion broadening levels, impuritioes affecting energy levels, etc.) are going to add much larger effects than simply not having an infinite duration beam. The widths of lines and the factors affecting them were a major field of study (and engineering) in the early days of the laser.

As for the crystal ends – it’s not just diamonds, of course, but all crystals. Most things aren’t single crystal, but polycrystalline. And no crystals (except perhaps a very few “whiskers”) are perfect even in their small extent. Crystal boundaries are, indeed, full of bonds that don’t join to other equivalent bonds. They are regions of high activity, and you can often make them visible by chemical action, like the etched surfaces on a meteorite. Chenical bonds don’t like to be unstaisfied, so you’ll find some bonding between different crystal elements, and some impurities attached there.

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[I’m going to ignore quantum issues here, and focus only on wave theory]. First, on the theoretical level, prisms don’t transform white light into multiple colored lights: they just deflect different colors at slightly different angles, and white light is just a bunch of colors superimposed. So the colors coming out of the prism will be exactly as well defined as the colors in the original beam.

And on the practical level, as Chronos said, ‘very long’ in this case means long in relation to the wavelength. I’ll add that I think the key length is the length of the pulse of light, rather than the length of the prism, but any real pulse of light is still plenty long: even a one centimeter long pulse of light has 20,000 wavelengths, which defines a color pretty closely. And note that to get a one-centimeter long pulse of light, you’d have to have an emitter that turns on and off in 30 picoseconds, which you’re not going to get by waving your hand in front of a beam of sunlight.

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Strictly speaking, no natural diamond is pure carbon; any naturally formed diamond will have some amount of impurities that are not evenly distributed through the stone. This heterogeneity is one way to distinguish natural diamond from artificially produced stones. There are, of course, free bonds at the surface features of a diamond (or indeed anything that isn’t a noble gas), and the particular character of the diamond allotrope of carbon gives its identifiable interaction with water droplets.

The light coming out of a prism (or as I see upon review, a laser) isn’t perfectly monochromatic or (in the case of a laser, coherent), of course, but given the short wavelength of visible light the necessary length of a “very long beam” is much shorter than what would be emitted in the time it takes you to flight a switch on and off. As CalMeacham notes, the impurities and sources of environmental noise with vastly exceed error arising from initial and terminal conditions.

Stranger

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Can you elaborate on this? I don’t understand.

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a true monochromatic light wave, if you analyzed it by performing a Fourier decomposition, would have precisely one component – a sine wave at that one frequency, but one which was always on, forever and ever. If you want to have a sine wave that is finitely bound in time or space – one with a bneginning and an end – you have to have components at other frequencies. These, if you analyze it mathematically, will turn out to be a spread of frequencies very close to that ifeal frequency, but with a finite spread. Conclusion: ant realy wave that has a starting time (even if it’s turned on gradually, and not suddenly) will not be truly monochromatic. It’s just the way it is.
You can say, “Oh, I just meant that it’s monochromatic only when it’s on”, but you’re trying to weasel your way out of an uncomfortable truth – the world goes on before it’s on and after it’s off, so you have to account for that time, too.

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I understand Fourier, but I still don’t get it. If a car is going 60 mph, we say that car is going 60 mph. At that point in time, surely it’s going that velocity. Why can’t we pick some instant in time, say right… now, and use that instant or some discrete time period to measure during. Just about everything else we measure with respect to time is done for some limited period.

Is this just the long way of saying that the wave won’t be monochromatic during its warm up and cool down periods?

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No. It’s because a true sine wave has neither a biginning nor an end. If you want to have, instead, a wave that looks like a sine wave that starts at some particular time and ends at anothere time (or which just starts at one time and never ends, or gradually starts up and gradually ends, or does anything besides just being on at the same intensity forever), then you have to include other frequency components. Ain’t no way out of it.
yeah, I can say whatever I want. But if I want to mathematically construct a wave that does this (or physically construct it, assuming I can get wave components that match the mathematic requirements without any other sort of broadening, and are all on forever, and have infinte coherence length), then this is the way to do it. It’s not a very physical situation, but it’s mathematically correct.

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Just adding what I know.

  1. From my P-chem buddies I hear that when you get down to femtosecond pulses artifacts like this start to show up. I have no idea how this would effect the “appearance” (in quotes since you probably can’t actually see a femtosecond pulse.) on the spectrum of white light except that any distinct bands would be broadened.

  2. Definitely there are no dangling carbons, at least not for long after you cleave it. Carbon anions, cations and radicals are extremely high energy species. Most likely they will react with oxygen, carbon dioxide and water in the air to form terminal hydrogens, hydroxyls and acids. This is also the case with the edges and defects in carbon nanotubes. With diamond you will probably end up with some unsaturated bits as well.

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About the prism: No matter how big the prism, if the white light pulse is really short, even though the emerging colored beams are of finite length they will be longer than the length of the pulse, and therefore be in existence before and/or after the pulse is in the prism.
About the diamond: Suppose you cleave a diamond into two pieces inside an evacuated chamber. There is nothing there to combine with the newly exposed carbon atoms. What do you have? A surface layer of unpaired electrons?

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What makes you say this? Unless your prism is phosphorescent or something (which is an entirely separate issue), the outgoing beams will be of the same length as the incoming beam.

I’ll bet that, in practice, you’ll end up with surface contamination from the knife that cut it, if nothing else. But whatever happens (contamination from unexpected sources, unpaired electrons, double bonds between surface carbon atoms, whatever), it probably won’t be too interesting, since it’s only affecting the atoms at the surface, which are a very small minority of them.

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You guys keep missing the point. These are thought experiments, not proposals for real operations.

The prism is performing a Fourier analysis on a finite wavepulse. Mathematically you get a superposition of infinitely long sine waves. They are infinitely long in time as well as in space. So why are no waves present in front and in back of the prism before and after the time the pulse was inside the prism?

Pretend you cut a perfect diamond with a perfectly clean knife in a perfect vacuum. What happens? Don’t give irrelevant details of why it can’t be done.

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The mathematical treatment idealizes the situation; in reality, the situation is more messy. The perturbative approach of quantum electrodynamics will, of course, agree that there are, in fact, fields in front of and behind the prism and all the nasty business in the transition is canceled out by destructive interference, degenerating to the same essential result as classical optics.

The two cut halves can then be stuck back together with no indication that a cut was ever made. Provide, of course, that you didn’t knock anything out of the matrix and you line everything back up as it was. Prior to that, you have a bunch of free bonds that are grabbing like crazy at anything they can find, like sorority girls in their final semester.

Stranger

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Don’t tell professor Hamers at the University of Wisconsin at Madison that. Sometimes surface functionality is all you need to do something interesting.

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There is a common simplifying assumption in the precisions with which you can do something or other with light of some wavelength, which is: You are distinguishing between the crest of a wave and the trough of a wave, or between the crest and the next crest.

This is the idea behind having telescope mirrors with better than quarter wave accuracy over their face. Traveling to and from the face, different rays of light, striking different areas that have a quarter wavelength difference in their location, will eventually be reunited half a wavelength out of phase, and will cancel rather than add. The diffraction limit imposed by having a half wavelength difference across the diameter of a telescope works the same way, and it is often said that it’s impossible to resolve smaller differences with an objective lens of that size.

But this is an unnecessarily restrictive qualification.

You can distinguish the directions to different radiobeacons using a loop antenna, to something like a degree of arc, even though the loop antenna is 1/10 m long and the wavelength could be kilometers long.

Similarly, you can describe a sine function over some limited domain, and you could also regress an expression of the correct form, with fantastic precision, even if the domain is a small part of one wavelength long.

I think it is correct to say that a sinusoidally varying function of time can exist for much less than a cycle, to say nothing of an infinity of cycles, at a frequency whose exactness is arbitrarily great.

Fourier transforms are a little funny this way. A fourier transformed dataset will become the amplitudes of the sine and cosine waves that represent zero, and one, and two, and three, and successive integer numbers, of cycles per the length of the dataset. NO frequencies that are noninteger multiples of one cycle per dataset are sampled. But the Fourier transform is just one particular mathematical transform, limited in its intent to considering situations that inherently have no beginning and end, such as the temperature profile around an iron ring after it has been wholly heated and then plunged halfway into icewater and left insulated in a pile of straw. This was the exact problem that Fourier invented his transform to study - the conduction of heat and resulting evolution of temperature profiles - according to what we now call Fourier’s law.

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Thanks to everyone who responded to my two puzzles. Of course I don’t believe that colored light will emanate forever from a prism. But there seemed to be a contradiction between two dogmas religously taught in beginning physics courses. a) a dispersive medium can separate any complex waveform into its constituent sinusoidal waves, and b) those sinusoids are by definition infinitly long. Most of the elementary physics we are taught refers to ideal cases which couldn’t exist in the real world. That is necessary, of course because the messy details would be too much for beginning students, but the amazing thing is that no one ever thinks to ask about, and no teacher ever even acknowledges, what seems to me to be obvious problems with these explanations.

Here are (I think) the boring details about the prism: When the white light first enters the prism there is no dispersion. The light passes through unchanged. It takes time for the electrons in the glass to pick up speed and start oscillating in time with the sine waves in the white light. This is what causes the dispersion. The emerging white light gradually starts separating into individual beams of different frequencies. When the white light is turned off the process reverses as the electron oscillations die out. Engineers call these things “transients.” The “monochromatic” beams are finite and fuzzy at the ends.

About the diamond: Probably right after the cleavage the diamond is covered with a monolayer of carbon radicals which immediately begin to rearrange in some way. Since the advent of materials science and nanotechnology these things are recieving much attention. but in my day everyone just assumed all covalent crystals were infinitely large, and carbon atoms were either isolated in space or combined in single molecules as s, sp, sp2 or sp3 hybrids. Again, how come we never objected to such an obviously flawed description?

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I wouldn’t discount carbanions and carbocations, but those would likely be in the minority. While some of them will rearrange in some way, most of them will react with the atmosphere. Perhapse you were including reaction with the atmosphere when you wrote “rearrange”, but chemistry wise they are distinct processes.

This thread would have been better titled “One physics question and one chemistry question”.

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All physics (and by extension, all natural sciences) are an attempt to describe complex and messy real world phenomena in discrete, categorically rigorous, and in the case of physics, strictly mathematical analogues. In order to do this we first have to abstract the problem so as to create some kind of boundary between the specific phenomenon of interest and its interactions with the rest of the world, and then find some mechanism which describes and predicts the behavior to a limit of measurement, or at least accurate enough for our immediate needs. For any real problem these interactions with reality are highly complex and too involved to teach as a gestalt to someone who is just being introduced to basic concepts. So in the beginning, at least, a teacher introduces a new phenomenon or mechanism by talking about it as an abstraction, and then once the basic principles are addressed, expands upon it.

Take, for instance, the case of work (in the strict technical sense). Work is fundamentally the effort done upon a mass to raise its energy potential. It is typically introduced in basic mechanics as moving a big blue mass m up a distance h, giving the resultant work product W=mgh, where g is acceleration due to gravity and W is the work (energy) done on the mass, equal to the increase in gravitational potential energy the mass now enjoys. The student is then told that the mass can now do work on something else equal to the work done upon it; that is, if we put the mass on a pulley or lever it can do an amount of work w on some other object as it returns to the ground state.

This description, of course, misses the entire business of loss due to friction, acoustics, the second law of thermodynamics, momentum dynamics, et cetera, and furthermore avoids the entire question of what this “gravity” business–the disturbing and mysterious action at a distance–is about anyway. But if you tried to go into the holistic explanation of all of this you’d never get anywhere and the student would give up in frustration. (Many do anyway, no matter how simple you make it.) But once you’ve got the concept of work down, then you can extend the concept to address the reality of work and its inherent connection to all motion and transfer of energy. Then work isn’t just about moving blocks up and down; it’s about expanding gas driving a piston, diffusion through membranes from ionic potentials, binding energy between nucleons, and all other physical phenomena involving energy exchange.

Of course, even once we’ve learned of greater complexities, we still typically return to a simplified concept of phenomena in order to solve actual problems, because trying to model every possible contribution is not only impossible but unnecessary. When I model the dynamics of a vehicle suspension of a car I don’t model the motion of the passengers or movements of the engine, because their contribution is too small to affect the results to within my calculation error. I might even leave out things that part of the primary system (like deformation of the chassis) because it is too complicated and not important enough to model accurately, or abstract other aspects (like tire squash, slew, and slip) will be replaced by a semi-empirical or regime-dependent response submodel.

In the case of the light entering and exiting a prism, the application of a Fourier series to describe the behavior of light is a simple and applicable abstraction, and very accurate in terms of getting a usable result as long as you are not interested in the specifics of how the beam enters and exits the prism. If you do want to delve into those specifics you need to get down into quantum field theory, which is beyond the scope of introduction to classical optics, or even basic modern optics.

Where the problem comes in is not that the explanation is incomplete–this is fundamentally true of all sciences, which always aspire to greater refinement and understanding–but that it is implicitly presented as a complete explanation without a nod to the greater complexity of real interactions.

Stranger

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I agree with stranger that approximations must be made and small perturbations must be left out of the phenomenon in order to make the explanation comprehensible to the student, but this is different. The model of the prism given to the student was not only a simplification, but contained an internal contradiction. The student needed to be told that a prism separates the colors in a light beam only after it has achieved a steady state. The explanation for dispersion is only valid if one assumes the white beam has been passing through the prism for a time that is astronomically longer than any of the frequencies in the light beam.

 On the other hand, many of the models used in science are self-contradictory at first. It is the contradiction that spurs us to find improved models. Consider Bohrs atom in the Old Quantum theory. Bohr said the hydrogen atom could only exist in discrete "stationary states," by which he meant states that could not change with time. Only this way could he explain why the electron did not lose energy and spiral into the nucleus. Then he immediately contracted himself by saying spectral lines were caused by the atom jumping from one of these eternal, unchangeable states to another. This contradiction spurred Schrodinger to translate "stationary state" into "eigenfunction of a wave equation". Although vibrating systems have only a discrete set of stationary states (normal modes) they can also exist in superpositions of these stationary states, which can change with time. Thus Bohrs atom can smoothly change fron one stationary state to another by changing the amount of each stationary state in the superposition of the upper and lower stare. The stationary states were demoted to states the atom existed in only when not being subjected to external influences, just as a violin string stayed in one or another normal mode only when not being fretted or bowed. By adding superimposed states to the Bohr theory and demoting the stationary states to a subset of all the states possible to the atom, the contradiction disappeared.