Quantum Physics

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Re: Quantum Physics

Post by halfwise on Thu May 10, 2012 7:58 pm

On the contrary I think it's nearly perfect. The concepts we are dealing with have aspects of both the waves and the particles we are familiar with. The only danger is that it might be such a strongly suggestive term that it might cause people to believe too strongly in the merge, thereby missing other aspects that don't fall inside the subspace described by 'particle+wave'.

In fact I know of at least one concept we'll get to in the 4rth installment that wouldn't really fit under either one.

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Re: Quantum Physics

Post by Pettytyrant101 on Thu May 10, 2012 8:07 pm

Thats kind of part of my gripe with wavicles. It suggests they know whats going on and its just a wave combined with a particle. When in fact its a WTF.
I just get the feeling scientists dont like coming out saying it is a WTF so they give it a vaguely fancy scientific sounding name instead that theyve cobbled together on the back of a napkin whilst on lunch.

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Re: Quantum Physics

Post by halfwise on Thu May 10, 2012 8:14 pm

Actually they are quite open about it being a WTF - nearly every quote from a famous scientist about quantum mechanics is variations on the theme of WTF. But at some point you have to get down to business, at which point it's useful to have some terminology.

In the case of wavicle it's not actually used much because everyone already knows that once you get to the small scale things go cockeyed, and we are always talking about electrons, photons etc which already have names. But to be honest, the term is usually in the back of my mind, reminding me to neither take the wave or particle properties too seriously, or more often to remind me not to just stick with one or the other.

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Re: Quantum Physics

Post by Orwell on Thu May 10, 2012 11:43 pm

Instead of "wavicle", why not "God"? Same element (sic) of doubt, and no actual proof. Very Happy

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Re: Quantum Physics

Post by David H on Fri May 11, 2012 12:32 am

...or Odo by the same logic Smile
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Re: Quantum Physics

Post by Pettytyrant101 on Fri May 11, 2012 12:33 am

lol!

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Re: Quantum Physics

Post by odo banks on Fri May 11, 2012 1:39 am

I have no idea what you mean --- and I don't like it! Mad

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Re: Quantum Physics

Post by halfwise on Sun May 20, 2012 7:19 pm

Well, since most people at the picnic appear to have collapsed into buckyfied heaps by now, I thought I may as well take this time to post the third installment of Quantum Mechanics.

NOTE: all the diagrams have disappeared in this installment. At some point I'll have to figure out how to fix that.


Part III:  Basic Principles

The way quantum mechanics works is actually pretty simple and clear-cut, so long as you are willing to let go your intuitive ideas of how the world works.  Most of what I present below closely follows Volume 3 Chapter 1 of the Feynman Lectures on Physics, even to the inclusion of his figures. Other than trimming his presentation down somewhat the major difference is that for many cases I can now include images from the actual experiments….just in case you don’t trust me.

Behavior of Particles and Waves

We return to that old chestnut, the double slit experiment.  Let us remind ourselves what would happen if we fire a stream of particles (bullets) at the two slits, with the proviso that some of these may bounce off the sides of the slits and so be deflected out of the straight stream (note that if the slit is smaller there’s a greater chance of the bullets hitting the edges, so the whole pattern would spread out more):


Now let’s repeat this experiment with waves:


If either slit is open separately, we see the same intensity of light on the screen, very reminiscent of what we’d see with bullets/particles.  This is why Newton was content to thing of light as particles.  But if we open both slits at once, we see the effects of interference – the final pattern is not the same as if you just added the results of each slit together.  The up and down motion of the waves is making itself felt by the interference pattern.  Since normally when you measure a wave at the high frequency of light you don’t notice all the ups and downs, just the amplitude of the wave, the intensity pattern doesn’t show the ripples.  But when the troughs and crests line up, the overall amplitude changes, and that is the pattern we see on the screen with both slits open.  When troughs line up with troughs, the amplitude becomes bigger.  When amplitude lines up with troughs, the amplitude gets smaller, and may cancel out completely.  Particles can’t do this…

Particle-Wave Duality

…or can they?  

By de Broglie’s principle, all particles have wavelengths that are inversely proportional to the momentum.  The objects we deal with on a daily basis are so large that even while sitting still the influence of vibrations from the floorboards will cause momentums so large that the wavelengths are miniscule, and therefore any interference patterns are also miniscule and undetectable.  But electrons are unimaginably small, so their small momentums produce wavelengths large enough to have noticeable effects.  We should therefore see an interference pattern if we use an electron gun instead of an ordinary gun shooting bullets:


The wave interference pattern tells you the probability of an electron hitting in any particular location!  If you look at the experiment with bullets, you’ll notice that you have a sort of bell shaped probability distribution.  That distribution is still there: note how the central peak is largest and then the peak heights on either side fall off.  The wave pattern is imposed on top of the particle pattern.  If the electrons are shot faster, the waves will become shorter, and the waves in the interference pattern will get closer together.  Once the waves become so short you can’t measure them anymore (either by making the electrons go faster or using more massive particles to increase the momentum) the interference pattern will become so tight you can’t distinguish the ripples, and it will look like you are shooting bullets again.  This tendency to go to the classical result at higher energies or masses is called the correspondence principle.  Basically every quantum result has to be able to look classical as you get further away from the lower energies.

In case all of this seems far fetched, here are pictures from an actual experiment done in the 1990’s.  The dots represent individual electrons, and they are allowed to pass through the slits one by one until through time the interference pattern is built up.


The result would be the same if you used photons: bit by bit the wave pattern is built up.  But the experiment has to be done very delicately to see one photon at a time, and until recent decades our technology was not sufficient to see individual photons.  Normally there are gazillions of them, so the interference pattern appears immediately, exactly as you would expect from the wave theory of light.

So this is the recipe for quantum mechanics: calculate the wavelength of particles based on their momentum (light also has a momentum calculated by dividing the energy by the speed of light), then create the interference pattern expected from these waves.  The intensity of the interference pattern at any point tells you the probability of measuring a particle at that point.  One thing you should remember from waves is that the intensity (energy content) of a wave goes as the square of the amplitude: we’ll see examples of that soon.

But wait a minute…if electrons are particles, shouldn’t they be going through one slit or the other?  If so, what are they interfering with?  It can’t be with other particles because in the experiment we just showed, the electrons can arrive one by one.  So a single electron must be acting like a wave going through both slits at once.  How can that be true?  Let’s set up an experiment to try to see what’s happening: we’ll put a little light source between the slits and watch for flashes of light that come from one slit or the other:


So we do this, and we see flashes of light coming from one slit or the other, but then a very discouraging thing happens: the interference patter disappears!  Once we force the electrons to act like particles, they do just that and we don’t see the wave effects anymore.   Nature won’t let us see both aspects at once.  This is called complementarity principle, either waves or particles.

It’s not necessarily mysterious, for when the photons bounce off the electrons they give them a little jolt, and even if they were forming an interference pattern, all these random jolts would smudge out the wave pattern.  Okay, so turn down the energy of the photons.  But when you do this, the wavelength gets larger, and you may remember that waves wash right around things that are much smaller than the wavelength.  If you do the math, you find that once the wavelength of the photon gets much larger than the distance between the slits, it gets harder to tell which slit the light is reflecting from and the interference pattern begins to reappear.  This is related to the famous uncertainty principle: as the kicks of momentum to the electrons get smaller, the distance within which you can pinpoint the location becomes larger.  We’ll come back to this later.

Particle in a Box

So we’ve seen how electrons and photons can act like ripples and interfere, but this isn’t exactly what’s happening in atoms.  In that case we’ve got waves that are constrained by the electric attraction between protons in the nucleus and electrons that are orbiting.  You may remember that when waves are constrained within boundaries they bounce back and forth and interfere so that only the right wavelengths will form standing waves.  It’s the same principle of interference, but the waves are hitting head-on rather than almost parallel. Since the ‘boundaries’ of an atom are defined by the inverse square law of electric attraction things can get a little complicated, so before we jump to that, let’s build up intuition by looking at a simpler case of electron confinement.  

We will look at the classic thought experiment known as the ‘particle in a box’ (yes, you can google it).  The idea is that the particle can move freely inside the box, but the walls are infinitely strong so the particle cannot penetrate the walls at all.  You can think of the strength of the walls as sort of being equivalent to the height of the walls: can the particle jump out?  In this case the particle can’t escape at all, so the wave amplitude and hence the probability (which goes as amplitude squared) must go to zero at the walls.


So for each harmonic we get another hump of probability density in the box: in the ground state we are more likely to find the particle in the middle.  Then we give it more energy and the wavelength increases, but it has to increase by the right jump so we get the next standing wave – and suddenly the particle won’t be found in the middle, just off to each side of the middle, and so on.  The possible energies are quantized, and the likely locations of the particle will change with each quantum state.  

It may seem like an extra step to first get the amplitudes and then square it to get the probabilities: why not say just add a hump for each energy level?  Well, that may work in this case, but it’s not so clear in other cases, such as seen in the two slit interference patterns above.  We need the positives and negatives of the waves in order to interfere and cancel out, otherwise we’d never get the locations of zero probability you see above.

A note on the correspondence principle: if this were a classical particle we’d be equally likely to find it throughout the box instead of for instance near the middle as found in the ground state.  But as we go to higher energy note how the humps become evenly distributed and at much higher energy the particle becomes equally likely to be found anywhere.

What if the sides of the box weren’t infinitely hard, but were kind of soft?  This is equivalent to having walls that are not infinitely high but have a defined height.  If the particle had enough energy then in the classical world it could bounce out, but if the energy was too low it would be stuck just as much as if the walls were infinitely high.  What happens in this case of soft walls is that some of the wave vibration spills into the walls, and we get the following situation:


So the particle is sort of seeping out of the box, and the probality of finding it a short distance outside of the box is not zero.  It’s location is not well defined. If the walls were narrow enough, it would be able to ‘tunnel’ out and escape.
This is equivalent to the ‘box’ that a proton forms for an electron: it’s not a well defined rigid wall, so the probability of finding the electron further away from the atom sort of fades away to zero.  If you were looking for the location of an electron around the nucleus, it would look like this:


where you see the ground state at the bottom, and various higher energy states above it.  You can see why it is often referred to as the ‘electron cloud’.  Still a particle in a box, just a more complicated box.

The Heisenberg Uncertainty Principle

Because particles behave as waves (until you disturb them by trying to measure the situation) there’s a certain fuzziness associated with them.  Attempts to make precise measurements come at a price.  Heisenberg’s very abstract version of QM showed relationships between certain variables such that if you increase the accuracy in observing one, the other responds by becoming less accurate.  Let’s look at this phenenoma in several settings to get a feel for it.

The diagram above shows a section of a wave moving past.  The wave also represents a particle, and the fact that it is just a section of a wave shows that the particle is ‘localized’ within a space Δx – it cannot be found just anywhere, but only where the wave has non-zero amplitude (you find yourself using phrases like ‘non-zero’ a lot in physics).  We often call such a wave segment a ‘wave packet’.  So Δx is the uncertainty in particle location.  If you want to know the momentum, you must measure the wavelength.  You do this by counting the number of wave crests n and dividing it into Δx: wavelength = Δx/n.   Remember, in the real world you don’t have the benefit of a nice diagram like this, so you have to look at things you can easily measure like the crests of waves that come by.  In this case since Δx is two wavelengths, you’d get

Measured Wavelength = Δx/n = (2 wavelengths)/(2 crests) = 1 true wavelength

It all seems so simple.  But wait: what if the dotted portion of the wave packet was missing?  Then Δx = 1.5 wavelengths, but we’d still measure 2 crests, and get a wrong wavelength:

Measured Wavelength = (1.5 wavelengths)/(2 crests) = 0.75 true wavelengths

So there’s an uncertainty in wavelength related to the uncertainty in position.  What if the length of the wave packet was about 10 crests long?  There would still be a possibility of about ½ of a wavelength not being measured.  We might get

Measured Wavelength = (9.5 wavelengths)/(10 crests) = 0.95 true wavelengths

So as the uncertainty in position Δx goes up, the uncertainty in measuring wavelength goes down.  This is a standard measurement technique: you measure many things at once and divide to reduce the measurement error.
Wavelength is related to momentum, so a greater uncertainty in position means less uncertainty in momentum.  Since momentum is often indicated by the symbol p, this is often written as

ΔxΔp > h

Where h is…..wouldn’t you know it….the ubiquitous Planck’s constant.  If you measure position more accurately, you get a less accurate measurement of momentum.  The ‘greater than’ sign ‘>’ is there simply because it’s always possible to make a worse measurement.  Heisenberg’s uncertainty principle tells you the limits of the best measurement.  

It can also be related to energy and time.  As the wave packet becomes shorter and the ability to measure wavelength accurately becomes worse, this means the ability to measure energy becomes worse.  But a shorter packet will travel past in a shorter time, so you get a greater accuracy in specifying the exact time the particle flew past.   This can be written as

ΔtΔE > h

And is just another version of the uncertainty principle.  This version has been used to assume that particles of mass E/c^2 (Einstein’s formula) can appear out of nowhere for a time t, so long as E•t < h.  This is called the ‘vacuum energy’ and has been confirmed by the ability to detect charge separation in what should be empty space.  Put two metal plates very close together, and if positive and negative charged particles pop out of nowhere (charge must be conserved, even if mass does not!), one plate may capture a positive while the other captures the negative, and there is a force pulling them together.  This is called the Casimir effect, and measurements conform to theory.  Kind of mind blowing, and it all comes from taking the uncertainty principle so seriously you risk being laughed at – until somebody does the experiment and the scientific world is turned on its collective ear.

So that’s the theory, how does it apply to various situations?  

We start by shooting particles (electrons, photons, what have you) through a slit of width Δx.  Because they behave as waves, they spread out instead of going straight through.  But how much do they spread?  As particles they start with a momentum we’ll call P.

The position is restricted in the vertical direction by an amount Δx.  In response the particle can pick up any random amount of vertical momentum with an average value ΔP = h/Δx.   If we make Δx smaller, ΔP will become bigger, thus spreading out the spray of particles, just as happens if the slit that a wave passes through is made smaller (review section 1).  The process can be reversed:  you should recall that if the wavelength is made shorter, the transmitted wave spreads out less.  This is done by increasing the momentum – so as particles become more energetic the spreading due to wave effects decreases and we approach the classical world.

Our other example comes from the particle in a box, for which the wavelength is fixed by the size of the box:

Note how as the size of the box ΔX goes down, the wavelength decreases as well.  Now momentum increases as wavelength decreases, so the momentum in the second box goes down.  At this point you may object that the momentum is completely defined by the wavelength, and since the wavelength is very well defined the momentum is too.  You’d be right, except for one small thing: for standing waves the waves are moving both to the right and to the left, and you don’t know which way!  So this provides an uncertainty in momentum, and as the size of the momentum goes up, so does this uncertainty (before you may not know, for instance whether the particle is moving 1 m/s to the right or the left, then you make the box smaller and you don’t know whether it’s going 2 m/s right or left).  The uncertainty in position is of course the size of the box itself, for the particle can be anywhere in the box with various probabilities set by the wave amplitude at each point.

This particle in a box situation can actuallly be used to estimate the size of atoms from the lowerst energy they emit: the energy suggests the momentum, which can be used to infer the size.

I’ve described the obvious cases, there are many more subtle ones that Einstein used to pose to Bohr.  Einstein hated the uncertainty principle, saying ‘God does not play dice with the universe.’  So he’d send a situation to Bohr, who would often spend a sleepless night turning it over in his mind before he found nature’s escape plan to keep us from knowing her too intimately.  One case used photons getting trapped in a box, and Bohr had to solve it using Einstein’s own equations of relativity, neatly leaving Einstein with no escape.  Since the details would require me to go into relativity (another set of treatises perhaps) I will not reproduce it here, but can’t resist a quote  from a physicist (Leon Rosenfeld) who was at the conference and described the scene:

“It was a real shock for Bohr...who, at first, could not think of a solution. For the entire evening he was extremely agitated, and he continued passing from one scientist to another, seeking to persuade them that it could not be the case, that it would have been the end of physics if Einstein were right; but he couldn't come up with any way to resolve the paradox. I will never forget the image of the two antagonists as they left the club: Einstein, with his tall and commanding figure, who walked tranquilly, with a mildly ironic smile, and Bohr who trotted along beside him, full of excitement...The morning after saw the triumph of Bohr.”

The decades long debate between the two giants is one of the humanizing sagas of Quantum Mechanics.

Seeing Ghosts

The core of Quantum Mechanics has been laid out.  To draw this section to a close we first summarize the core ideas, then illustrate them with an experiment that not only clarifies QM, but brings out some of the strangeness (as if it were not strange enough!) that will be investigated in the final section.

1. Any particle should be treated as a wave with wavelength WL given by de Broglie’s postulate WL = h/momentum.   This wave propogates in the direction of the particle’s momentum, but will spread out through space and interfere with itself even if you are working with a single particle.
2. Single particles are detected with a probability density at any point given by the amplitude of the wave squared at that point.   As many particles are detected the wave pattern is built up by the density of observations.
3. If anything is done to localize the particle at any point in the wave, the wave pattern is destroyed (the wave function ‘collapses’ to a point upon observation).
4. The wave nature of particles means that as the position of a particle becomes more accurately localized, the momentum becomes more uncertain, and the reverse.  This relationship can be converted into energy and time.

So as a final experiment we look at what happens if you split a beam of light in two, send it through widely separated paths and then let it come back together and interfere with itself.  The splitting is done by use of a half-silvered mirror, so that half the light is reflected and half the light transmitted.  Other mirrors are used to guide the light around it’s two tracks.  In the diagram below, light blue is the half silvered mirror, dark blue is the full silvered mirrors, light is shown in red, and all other physical objects that don’t transmit or reflect light are shown in black.


The light splits into two paths.  If the paths are equal, both beams of light just recombine at the detector as though nothing had happened.  But if they are not equal, what happens on recombination depends on what fraction of a wavelength difference there is between the paths.  If the difference is a whole number of wavelengths, crest will line up with crest and the full intensity of light will appear at the detector.  If half a wavelength, or one and a half wavelengths, etc, the light will cancel out and nothing will be seen at the detector.  (It’s a fair question to ask where the energy goes in that case, and the answer is that light is a wave that always spreads out, and you’ll see stronger intensities in areas other than where the detector is).

In the diagram drawn it looks like the two paths must be the same length, but we can make the situation not quite rectangular, and we only need to deal with a few wavelengths of light to see the interference between the waves.  We can rotate one of the mirrors slightly to skew the difference, and at the detector we’d see alternate light and dark as the distance changes.  If we swing the door shut the top path, the interference pattern would disappear and we wouldn’t see the alternating light and dark anymore.

That’s the classical picture.  In the quantum picture we know we only detect one photon at a time, and we have instruments that prove it – giving off clicks at single points in space and time.  Let’s say we turn the light down so low that only one photon at a time is going through.  We could put a detector at the beginning and only see it click every second or so (but of course it wouldn’t go through the experiment), we could put a detector at the end and it only clicks every second or so.  We could put the detector in either of the two paths, and it would only click half as often as at either end of the experiment.  

Clearly only one photon is doing one part or the other, but by changing the path differences we see the light and dark patterns that tells us that each photon is obeying the interference law that depends on a wave going through both paths.  We can set the door on the upper path to remain open until the time the photon on the top path is about to pass, then swing it close just long enough to stop it and swing it open again.  The interference pattern will vanish.  But how does the photon on the lower path know there’s not supposed to be an interference pattern?  The door just flickered shut, how did this information get communicated?  If the photon is a real entity travelling through the experiment, it’s a long ways from the door, and it’s very possible that information about the door being open or shut would have to travel faster than light to inform the photon whether it has to obey interference or not.  Either that or you have to believe there is a wave that travels all through space and then randomly turns into a photon at the end.  This is the “spooky action at a distance” Einstein was complaining about.

I’m sorry to be the bearer of bad news, but at this time there is still no resolution to this problem.  For the most part there doesn’t have to be a resolution: for decades physicists have been dealing with waves that magically turn into particles without worrying about the implications because it works.  Why mess that up by thinking too hard about it?  But of course, we can’t just leave it like that.  Many have thought about it, and that’s the topic of the final installment.


Last edited by halfwise on Mon Apr 13, 2015 2:38 pm; edited 4 times in total

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Re: Quantum Physics

Post by Mrs Figg on Sun May 20, 2012 8:08 pm

is it ghosties? but not real ghosties, ghosties that were and are at the same time.
particles that shouldnt be there but they are. Shocked

(((sorry thats my inane question after all your hard work, I am still digesting most of it, and its not really a question as the answer would be scary)))))))

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Re: Quantum Physics

Post by halfwise on Sun May 20, 2012 8:25 pm

the waves do seem to be everywhere....in fact in the next installment I'll talk about how you have to treat the waves as going through all the universe in order to get accurate answers that agree with experiment. The particles just appear when you jiggle the waves by trying to observe them. So yeah, the particles seem to be these mysterious ghosties that are everywhere at once, though the maximum probability is usually pretty close to where you'd expect the particles to be in any circumstance.

No, it doesn't make complete sense. The only reason anyone accepts this stuff is because it seems to work really, really well. I'm talking multiple decimal point accuracy. Nobody asked for the world to be this way....

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Re: Quantum Physics

Post by Mrs Figg on Sun May 20, 2012 8:41 pm

Is it still true that the particle went faster than the speed of light in the big tunnel thingy, (cant remember the name) or have they discounted it? I remember reading they were re-looking into it.

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Re: Quantum Physics

Post by halfwise on Sun May 20, 2012 8:47 pm

discounted. Loose cable.

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Re: Quantum Physics

Post by Mrs Figg on Sun May 20, 2012 8:48 pm

No really? how disappointing. But no really really?? No

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Re: Quantum Physics

Post by halfwise on Sun May 20, 2012 8:59 pm

Relativity fits together so beautifully and explains so many things that most physicists were convinced it was an error. They only published because they couldn't find the error and so were at wit's end. The team leader has since announced early retirement.

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Re: Quantum Physics

Post by Mrs Figg on Sun May 20, 2012 10:02 pm

maybe there is no error? I really hope its true, that means there is room for the weird and impossible in our scientific certainty. It means everything we know is wrong.

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Re: Quantum Physics

Post by halfwise on Sun May 20, 2012 10:16 pm

I'd place more hope in dark matter/energy than relativity.

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Re: Quantum Physics

Post by Pettytyrant101 on Sun May 20, 2012 10:17 pm

Things appear to be able to be in two places at once and can communicate across any distance instantly! Seems plenty bloody weird enough already!

ps another excellent, informative piece Halfwise (that will again take me several reads to begin to fully digest).

I remember the head writer on Star Trek: NG being asked how the Hiesneberg compensator worked on the transporter. He replied, 'very well thankyou.' Very Happy

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Re: Quantum Physics

Post by Mrs Figg on Sun May 20, 2012 10:33 pm

cheers good answer. I wonder if spooky whotsit at a distance is like the transporter? for a split second you can be wibbly wobbly in two places at once. Very Happy

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Re: Quantum Physics

Post by Pettytyrant101 on Sun May 20, 2012 10:35 pm

I often get the sensation I'm wibbly-wobbly in two places at once- then the waveform collaspses and my head hits the floor. drunken

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Re: Quantum Physics

Post by halfwise on Sun May 20, 2012 10:35 pm

There was another cool incident like that in an episode that used "quantum filiments".

Low rank Engineer: "Are those anything like cosmic strings, sir?"
Geordie: "No."

end of scientific exposition.

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Re: Quantum Physics

Post by Pettytyrant101 on Sun May 20, 2012 10:41 pm

Not to go off on a tangental rant but I will anyway cause I'm me- but the fact the ST tv series mentioned cosmic strings and hiesenberg compensators nicely and neatly demonstrates why it was good and the ST reboot films are shit!

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Re: Quantum Physics

Post by Norc on Sun May 20, 2012 10:43 pm

this thread. It puzzles me. Someone here is actually trying to teach us something?
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Re: Quantum Physics

Post by Pettytyrant101 on Sun May 20, 2012 10:46 pm

What about my thread on religion too?! Mad Youth today! Laughing I dont know why we bother trying to help them Halfwise, better off just staying drunk I reckon. Nod

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Re: Quantum Physics

Post by David H on Sun May 20, 2012 10:52 pm

Petty! Is that you? I'm coming for you.....
You can't hide, and when I find you there'll be hell to pay, no matter what your religious thread says! Extremely Crabbit
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Re: Quantum Physics

Post by Pettytyrant101 on Sun May 20, 2012 10:53 pm

{{{{{{I'm not here Sofa }}}}}}

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