Schrödinger's hat

If I had, for example, a ball on the end of a spring. I pull it back and let it go, then have a look a little while afterwards. The simplest approximation for where it will be in a small amount of time is its current position. [math] x(t) = x_0 [/math] This isn't very good, but for a little while, it'll be about where it is. This is a zeroth order approximation. Slightly better is a first order. I look at how fast it's going and add a little bit. [math]x(t) = x_0 + v_0 t [/math] This will give us okay answers for slightly longer. But it's also accelerating because of the spring: [math]x(t) = x_0 + v_0 t + \frac{a_0}{2} t^2[/math] (second order) and then the change in acceleration and so on. The nth order comes from the exponent in the time term. So if it's second order in time we go up to [math]t^2[/math] This works out well for [math]t<1[/math] in most cases because if you raise a small number to a high power you get an even smaller number. Do some reading about taylor series if you want to know a little more.

My logic was more along these lines: Simple equations pop up in nature easily Sequences related to exponentials can be generated by simple equations If we want to avoid being mistaken for a natural event, we should avoid such sequences Something else relevant: Pulsars emit radiation on various bands in an extremely regular manner. There was some speculation when the first radio pulsars were found that they might be some kind of signal. For something a bit more complicated/speculative, consider a pulsar that was losing rotational energy to its environment at a rate proportional to its speed -- something that happens in small scale phenomena, quite a lot. I don't know the exact situation for a real pulsar, and in our observations this process is very slow, but this is a plausible for the type of thing you'd see naturally (if not a specific example). It would be governed by the equation [math]\frac{d\omega}{dt} = A\omega[/math] Which has the solution [math]\omega = e^{At}[/math] As such the time between the pulses would be a geometric series.(something along the lines of 1 2 4 etc, but more likely 1, 1.0324, 1.0324^2, and so on) For anything more specific you'd probably have to talk to someone who does a lot of radio astronomy.

More that the probability would 'leak' out of the bound state (like energy leaks out of the string) and you'd find the electron somewhere else, such as in a free state (which does not have a discrete energy spectrum). It might then emit a photon and become bound again.

Humm, I'd never considered that aspect. How would you rate actionscript against logo in this role?

I..uhh. You're going to have to give me an example.

It's Newton's equations that got us to the moon, but it's GR that gets you to the shops and back without getting lost.

And space is a rubber sheet. Gotchtya.

The problem comes with english lacking good words for many concepts. If I had a langauge without words for engine, or metal or air I would find it very hard to describe what an aeroplane is in all but the grossest simplification.

I prefer to put it this way: (Up to some unit conversions/factors of c) Some energy-momentum is mass, all mass is energy-momentum, but not all energy-momentum is mass. The word momentum is used for either all energy-momentum, or the kind that is not mass, depending on context. Photons have the type of energy-momentum that is not mass. Also mass in this and similar contexts is rest mass (or invariant mass, as they are the both zero for a photon), not relativistic mass.

The gamma rays come from in front of the event horizon, not behind it. As matter falls into the black hole it gets compressed, heated up and pushed against other matter so much that the energy in it can be released (through nuclear reactions, and in extreme cases the matter will annihilate completely).

I did look into this once, trying to dredge up the knowledge from my memory. Part of it is that the sun is quite loud in shortwave/am frequencies. Also FM is quite directional, so you can get higher gain antennas more easily and/or align your receiving arial with the polarization of the transmitting one. But more significant has something to do with the ionization of the ionosphere. Edit: I recall after a bit of help from the internet. This goes into some detail http://radiojove.gsfc.nasa.gov/education/educ/radio/tran-rec/exerc/iono.htm TLDR version is the more free electrons there are, the more reflective the ionosphere is (free electrons are what makes metal shiny). During the day there is more ionizing radiation knocking electrons off of their atoms. This results in the atmosphere being more reflective. FM isn't effected because it travels fairly close to line of sight. AM and shortwave will bounce off of the ionosphere as they travel (which is why you can pick them up from much further away).

Entropy.

This is just an image. You can include images from a url or as attachments. I don't recognise the software Daedalus used for his graph, but one free package is octave, it's an open source version of matlab using the gnuplot package for graphics (you can also install other graphing packages). Graphics can be produced directly in latex. [latex] \setlength{\unitlength}{1mm} \begin{picture}(60, 40) \put(20,30){\circle{1}} \put(20,30){\circle{2}} \put(20,30){\circle{4}} \put(20,30){\circle{8}} \put(20,30){\circle{16}} \put(20,30){\circle{32}} \put(40,30){\circle{1}} \put(40,30){\circle{2}} \put(40,30){\circle{3}} \put(40,30){\circle{4}} \put(40,30){\circle{5}} \put(40,30){\circle{6}} \put(40,30){\circle{7}} \put(40,30){\circle{8}} \put(40,30){\circle{9}} \put(40,30){\circle{10}} \put(40,30){\circle{11}} \put(40,30){\circle{12}} \put(40,30){\circle{13}} \put(40,30){\circle{14}} \put(15,10){\circle*{1}} \put(20,10){\circle*{2}} \put(25,10){\circle*{3}} \put(30,10){\circle*{4}} \put(35,10){\circle*{5}} \end{picture} [/latex] (copied and pasted from http://en.wikibooks....eating_Graphics ) But plotting directly is cumbersome without extra packages which this forum doesn't seem to have installed. The forum also tends to strip out carriage returns from latex for some reason. This makes things a bit hard to read at times.

Serpinski triangle: Draw a triangle. In the middle of this, draw an upside down triangle, dividing it into four triangles Everywhere you see an upright triangle, draw an upside down triangle in it until you get bored Fern fractal's are also interesting: http://en.wikipedia....i/Barnsley_fern They are based on copying the full image into itself in four different ways

Hmm, energy loss and wave speed in a bathtub is too slow for this to make an intuitive model for explaining this. I also don't know how well a classical wave and energy loss translate to bound quantum states, but I'll give it a go. So I'll fall back on my favourite example: A guitar string, or other taught wire. You can feed energy in at the wrong frequency, but it tends to leak out again. If you think of plucking a string (or even better is rubbing it with a bow like a violin), what you're inputting is a big mixture of frequencies. You'll note that when you pluck the guitar string you don't hear white noise coming out again. This is because the energy of frequencies that aren't harmonics doesn't stay in the string very well. Instead it goes into moving the head of the guitar, or transfers to the body very quickly. You can see how this will be the case if you look at [math] v=f\lambda[/math]. When f is not a harmonic, the wave amplitude cannot be zero at the ends of the string, so it will wind up pulling on the ends where it is attached much more. One way to hear this is to mute the string with your other hand and pluck it. Then listen carefully while you pluck it normally. You'll notice there's the same sharp plucking sound. As a result of all this, the string (a while after being plucked) will only carry energy in frequencies that are harmonics of the base note. Mathematically these harmonics are equivalent to the bound states of our quantum particle. The other states are the ones that don't fit into the classical analogue very well. I usually think of it as a sum-over-paths thing (very much closer to your thoughts about the waves crashing into each other). Will ponder further. This could just be what Dr Rocket is always saying, 'Difficult questions have simple, easy to understand, wrong answers'.

The size of the various orbitals decreases with increasing number of protons. The shape stays pretty much the same, because you still have the same symmetries.

Indeed, and this is primarily what the first two (and some of the third) meditations are about. Establishing that there is some kind of thing generating this perception of thought, regardless of all other factors. Destartes speaks of knowing that he is a thinking being, but bear in mind that this is the/one of the first work(s) on solipsism. On top of that, We read it in a different language to the one it was written in. Because of this I think it's quite reasonable to allow some room for interpretation on the exact definition of 'me' and the distinction between thought and perception of thought. Getting bogged down in semantic issues around these two words serves no purpose I can think of in this discussion.

Well it'll change shape a bit, but still loosely resemble that from a bar magnet up close, and very closely resemble it from a distance.

You should never stop thinking. Just don't close your mind off to the (quite likely when anyone is speculating like this) possibility you may be wrong. Re. space being curved to make earth round: This is kinda what makes the event horizon to a black hole round, but space around earth is not curved enough to do this. We can detect curvature in space, even if we can't get to a higher dimension to look at it. If you imagine an ant on a balloon that can't see the third dimension. Let's name him Steve. From Steve's point of view, he's in a two dimensional space. He could try and figure out how curved the balloon/space is by going around in a loop and doing something called paralell transport. This is a bit hard to explain without a bunch of maths, but one way to look at it is like driving a car without turning the wheels. If you drive a car along one of those tracks with extremely banked corners it will turn even though the wheels are straight. First he drives forward. Then does a similar action to the right (without turning the car, or maybe has a car on a car pointing the other way, the analogy breaks down a bit :/) then backward, then left. If he is in flat space he winds up facing the direction he started but in curved space he'll wind up pointing in a slightly different direction. This turning is one way of measuring curvature, and in earth's gravity it isn't nearly enough to make the surface loop right around.

Very loosely, it'd have north pole all the way up the angled surface. Or you could model it as a bunch of magnets of different lenghts/strengths all side-by-side.

It proves that Descartes -- whatever he/it is -- exists. Again it comes down to issues of semantics. Iggy's definition of 'me' in Descartes' argument is broader than mine, which is broader than yours.

If you cut a magnet, you wind up with two smaller/weaker magnets. The best way to think of it in order to not get confused is this: Imagine the magnet being made of many tiny magnets. If they align the magnetic field adds up to a stronger field. So if you cut your big magnet, the tiny magnets still line up and add up. If you were to then rotate one of the pieces, some of the tiny magnets would produce the opposite field to the others, so the net effect would be much weaker in most places (not zero because the strength also depends on distance).

42

Awesome question. For one, degenerate matter won't always be at its maximum density/all be degenerate matter. So it might just act like very very very strong dense normal matter (your bat may not survive, depending on how you're containing this degenerate matter). The other possibility is that it's at exactly its maximum density. In this case two things might happen: First, you'll force some portion of it to overcome degeneracy pressure. So if it was electron degeneracy pressure it might become neutron degeneracy pressure, Neutron degeneracy pressure would enter a state I don't know about. This would be the same process a neutron star undergoes in the relevant kind of supernova. Whether any amount of this matter would become a black hole, I don't know. This new state could be stable, if -- say -- it was a sphere you were hitting. It'd be the straw that broke the camel's back, so to speak and the object would collapse into either a neutron star or a black hole (the resulting explosion would be unpleasant). Or it might be unstable. If your degenerate matter were somehow magically a different shape you might change some of it into a black hole which will immediately explode due to hawking radiation. In some weird circumstance (maybe your containment is really good, but not based on the objects self-gravity) it might just settle back into being degenerate matter again, I'm not quite sure.

My brain actually parsed the time derivative bit as E.. While I'm actually talking I may as well say a bit about how the wave equation entails shape. The electron behaves almost purely as a wave in this case. And it follows that big scary wave equation I wrote. As mississippichem said, there are different quantum numbers involved, and they define different solutions to the equation. It's a bit hard to explain for an electron (or a quantum system in general). So let's use a classical one. Hopefully you can get access to a guitar/bass guitar. A violin or other stringed instrument also works if you're good, but guitar is best. Just touch the string on the 12th fret, and pluck it near the sound hole (or on your violin play a harmonic, if you can do this you probably already know about harmonics and can just follow the guitar explanation). What you've done is cancel out some of the resonant modes of the string. You can release it with both hands and you'll hear that the pitch stays 1 octave above the root note. The vibration on the string is a wave with one 'quantum number' known as a harmonic. Different harmonics have a different number (not really quantum number, but analogous in the maths), and they have a different shape. One is the full string vibrating, one is still in the middle with two halves vibrating, then three stationary points, and so on. If this were a quantum system you'd be more likely to find the phonon/particle/whatever where the string is vibrating most and unlikely to find it where it isn't vibrating much. The other difference is you can put energy into all the harmonics at once. The electron is much the same, but it does sort of a funky 3D 'vibration' (the more 'vibration', the more likely you are to find it) The different quantum numbers define the 'still' (unlikely to find) and 'moving' (likely) points in much the same way as the harmonic defined the number of still points on the guitar string.