I was going to reply point by point, but I really must do something else with my Sunday.
"I am not aware of any relativistic QM theory. In fact, I believe that to be one of the things most actively sought in modern physics."
General Relativity is the problem. Resolving Quantum Mechanics and Special Relativity (i.e. everything except gravity) has been done. It's called Quantum Field Theory. (Specifically Quantum Electrodynamics, or QED, for all things relating to the Coloumb force.) A lot has been done since the 1920s. I'd be surprised if you could get any useful molecular results from the Schrodinger equation.
`Particle in a box` is a toy problem, designed to introduce the student to the idea of boundary conditions. Imagine a guitar string, fixed at both ends, and that's your physical model. The `wave` isn't electromagnetic, gravitational or the like; it is a description of probability amplitudes, the square of which corresponds to probability (in the generally used interpretation).
The particle doesn't bounce off the walls. It is spread out, and doesn't rattle around like a classical ball. (If it did, you could measure the recoil as it hit the walls!)
As for quantisation emerging: the postulates do not include discrete eigenvalues. Read them again. Preferably from a better text book. It only emerges when you apply quantum mechanics to a a specific situation.
"...2+2=5..." The maths of QM isn't as hard as for number theory. And is certainly a lot more interesting.
"Physics courses in college take the same path." [They state unjustified assumptions.]
Have you taken one? Depending on the university, students are introduced the the subject in a bit of a rush, and only later (often in an optional course) come back to carefully examine the postulates.
Sure, assumptions are necessary to simulate a real situation (like a molecule). Maybe it's not a clean theory. Or maybe we just don't know how to frame the maths to do it efficiently on a computer. Eitehr way, just because we have to simplfy it for a specific situation does not render the whole approach invalid.
I stand by my comment that you know too little (you don't need years of study) to say it's wrong. As far as I can tell, it's not classical, so it doen't make sense/feel right, so you conclude it must be wrong. A lot of work has gone in to trying to reporoduce results with classical mechanics only, and we'd all embrace it, if only it would work.
Agreed - Spin cannot be derived from the Schrodinger equation alone, but it emerges naturally in the relativistic theory; there are far fewer ad hoc additions to QM than you suggest.
The Schrodinger equation is incompatible with relativity. Schrodinger knew this, but was unable to fix it; we needed Paul Dirac for that. For understanding the fundamentals of QM, and trying seriously to punch holes in it, a cursory understanding of non-relativistic QM is woefully inadequate.
Quantisation ============
Quantisation emerges - it isn't postulated. Eigenvalues can be continuous.
Take the momentum operator (which, as it's normally defined, just uses a differential to extract the momentum from the equation describing a travelling wave psi = e^(i (k x - omega t)). This wave describes a particle moving in free space. Nowhere have we suggested k should take only certain values and, hence, for a free particle, momentum k is continuous. Only when boundary conditions (i.e. particle in a box, or in the field around an atomic nucleus) are imposed do things start to become discrete. That quantities are discrete _emerges_ from QM, and is not deliberately built in to it.
"The hamiltonian is... also a clearly quantized entity."; "time-independent Shrodinger equation defines energy levels to be discrete eigenvalues"
Not so. For the travelling wave psi = e^(i (k x - omega t)), in free space (i.e. V(r) = 0), H psi = (hbar k)^2 / (2 m) - there is no mention of (hbar k) being constrained to take one of a certain set of values - it is continuous. If you impose, say, V(r) = 0 for (|r| a/2), you have a `particle in a box` scenario. The Schrodinger equation has to be zero at r = (+/-)a/2 (this is from the maths, not an additional postulate) and you find there are only certain waves which satisfy this condition. THEN momentum (and hence energy) becomes quantised.
Uncertainty Principle =====================
As for the uncertainty principle... if things are decribed by waves, it is inescapable. See the Bandwidth Theorem. For the case of a wave, this just says that the shorter a wave in time, the less well known it's frequency. Imagine playing a 1kHz note for less than 1ms; you would be hard pressed to work out the frequency if you just had this short sample to work with.
This was part of the inspiration/justification for the uncertainty principle, and is certainly not the whole story.
Schrodinger Equation ====================
To clarify, the Schrodinger equation is H psi = i hbar (d/dt) psi - nothing more.
The Hamiltonian, H, can take many forms, but when it is first introduced it just uses the Classical Mechanics Hamiltonian and replaces the usual variabes (such as momentum) with the equivalent operators. More generally, it can be used to write down the total energy of two systems (such as an atom and a field) AND the interaction between them. This interaction term is used to describe emission/absorption of light by an atom, for example.
Bohr ====
As I stated, Bohr's approach is outdated; he had to make up many things to make a working model, but his model is relevant now only as a teaching aid. Incidently, it was the accellerating charges issue that made him propose there were certain `stable` orbits; something (like an electron around a nucleus or a planet around a star) in orbit is always accellerating towards the centre (i.e. nucleus or star) and hence, if it's charged (as an electron is), it should emit.
Slits =====
Double slits - try searching for this on Google Scholar or the arXiv.
We physcists do like to keep things as simple as possible, and experiments have been done with apparatus very closely resembling Young's double slits, and not just in crystalography. Be careful when you talk of a conspiracy; all published scientific work is open to review, and the experiments are repeated independently. Such accusations could mark
I haven't read your comments carefully yet (I intend to, but it's nearly the weekend!). My comments may be less than coherent (see earlier comment about weekend) but, for now...
Interesting point about matter being the thing that's quantised, and not the light. Pondering...
You may want to look at the derivation of the black body spectrum as evidence of photons. It _seems_ impossible to reproduce observations without quantising the light. (i.e. if you don't, the maths implies something like a hot coal radiates huge amounts of ultraviolet light, which it clearly doesn't) http://en.wikipedia.org/wiki/Ultraviolet_catastrop he
(I suppose this could be seen as a quantum effect in matter, not light, but if this is how light is made, it presumably suggests light is also quantised.)
Good point about energy & time -> amplitude & phase; I'll ponder that. Problem, as far as I recall, is that phase isn't easy to extract from a wavefunction (in the way that momentum, position etc can be extracted).
The double slit experiment works just as well with single electrons, and bigger things such as Carbon-60 molecules.
Quantum Mechanics does explain the orbitals of electrons in atoms. Very well, in fact. Bohr's first stab at a model has been outdated, and orbitals are generally calculated using the Schrodinger equation (where THAT comes from is another matter!) and the electrostatic potential of the nucleus. Things like spin and such are included, and the results agree with experiment. There's no `artificial' constraint on the types of orbitals allowed - they're just the solution to the equation in that potential.
The Bell inequalities really are worth the trouble. They provide some very compelling results contrary to hidden-variables and approachs with suggest the only real probability is our lack of knowledge. I'll refresh my memory of them if I get a chance. Unless something has been overlooked, probability really is a state of nature. I don't like it, but I can't find a way around it.
There are lots of alternative theories to QM, usually proposed by people with no formal training in the field. I don't know your background, but you'd better know your conventional QM if you want to convince someone else to try to fill in the gaps in your theory. There are many people with the ability to either shoot this down, or make it in to something complete. (I am not one of these people!) i.e. I know lots who would be capable, but it mightn't be easy to convince them to spend their time trying.
Some of Feynmann's interpretations seem odd, but his Lectures in Physics series is excellent. The conceptual issues part of Chris Isham's work http://www.worldscibooks.com/physics/p001.html is also excellent (I had the good fortune to hear him lecture, and I hope it comes across half as well when written down). He deals with all the maths first, but the Conceptual Issues chapter can stand alone.
First off let me qualify this response by saying I'm answering in the context of quantum mechanics. As we both agree, this certainly isn't a complete theory of how the universe works. However, there appear to be a few misconceptions which I hope to help clear up so that they don't cloud the real issue of the inadequacies of QM...
>> You can't make a perfect copy of an individual quantum (i.e. one photon). If you >> have an lot of quanta (i.e. loads of identical photons coming out of a laser)
>Those photons would be perfect copies of one another, wouldn't they? Ok, so the laser can't produce identical polarization, but it certainly can produce >identical phase and amplitude; that's what a laser beam is all about. Splitting a beam does not disturb those parameters and they can be separately >measured, as was done in the experiment. A measurement of the left beam would produce the same result as the measurement of the right beam, so you can >measure in one and know in the other. Phase and amplitude of a wave may be considered its equivalents of position and momentum, so doing this disproves the >uncertainty principle by example.
When talking about quantum mechanics involving photons, we usually consider their polarisation. This is because it is a variable which has only two components; it can be represented in a two-state basis[1]. When we make a measurement, we measure along one of these components. In the case of the beamsplitter I mentioned previously, we `split' the photon by sending the vertical component one way, and the horizonal component the other.
It isn't so neat for amplitude and phase. Firstly, they are not conjugate variables (like position and momentum), so measurement of one does not imply the other is disturbed. But let me come back to this...
Light polarised, say, linearly along a 45 degree axis to the vertical would have one component reflected and the other transmitted by the beamsplitter. This does suggest the amplitude of both outputs is equal (and so would the phase). If we use a mirror to shine these two outputs on to a screen, we will see interference. This result is consistent with waves, classical electromagnetism, common sense... there's no mention of quanta.
Unfortunately this does not correspond to duplicating the photon. Consider the photoelectric effect. http://en.wikipedia.org/wiki/Photoelectric_effect This is a good rugged experimental demonstration that light interacts in the discret lumps we call photons; the implication[2] is that it also only exists in these lumps. So, if we send in just one photon in to our beam splitter, which output does it come out of? It can't come out of both, but we see interference when we recombine them (and we don't if we block one of the paths).
What I've very awkwardly described is Young's Double Slit experiment. http://en.wikipedia.org/wiki/Double-slit_experimen t You probably know of this, and it's no mystery with waves; but it works with individual quanta too. (i.e. only one quantum in the aparatus at any one time, but interference is still seen!) So... which slit does the quanta go through?
If you're able to resolve the double slit experiment and the photoelectric effect without invoking superpositions, I'd be very interested to hear.[3]
[1] A basis is a set of states which we use to represent the value of some variable. For polarisation, there are several including: linear vertical & linear horizontal (i.e. the electric field of the light points up and down or left and right) and left circular & right circular (i.e. the electric field traces out a helix with one of the two possible helicities). These states are orthogonal in the same way as the x- and y- axes on a grid are; no amount of x can change the position in y.
[2] Maybe this is the stumbling block, but I'm sur
Now I've had my coffee, let me add to my last comment...
You can't make a perfect copy of an individual quantum (i.e. one photon). If you have an lot of quanta (i.e. loads of identical photons coming out of a laser) you can make copies, because you can measure lots of quanta and take the average to work out (to an accuracy limited only by the number of quanta you have) all the parameters you need to set up your equipment to make copies. This is all because you destroy (or at the very least disturb) each quanta when you measure it.
In the case of a photon, you'd probably (in principle) use a polarisation-sensitive beam-splitter and two detectors, one on each output from the beam-splitter. The photon's polarisation isn't just disturbed - the photon is absorbed and hence destroyed when it hits the detector and is measured. If (as would be the general case) it was in some superposition of both states, we'd still only measure either one state or the other, with the appropriate probability; you'd need lots of states to work out what this probability was.
In a laser, the photons it emits are all identical (in principle), but their polarisation is set by the geometry of the device or by spontaneous emission of photons from within the gain medium of the laser. Unfortunately, it's not enough to set the laser up, send in one arbitrarily polarised photon, and expect many copies of it to be emitted.
This isn't just a technical problem. A device which amplifies some signal from an individual quantum necessarily introduces noise, as do all amplifiers. This most you can get out of this device at the end is a classical probability, rather than the coefficients of a quantum superposition (complete with phase as well as amplitude).
Such a quantum amplifier would be needed if we were to actually make a macroscopic superposition, such as Schrodinger's famous cat in a box. The result of a radioactive isotope decaying or not, with 50% probability, would have to be amplified so that the result could be used to decide whether or not to break the beaker full of poison gas to kill the cat. Through the process of amplification, the quantum superposition is changed to a plain old classical probability, and the cat is either dead OR alive before we open the box - no observer required! (This tending towards a classical probability from a quantum one has a sound mathematical backing in the ugly but effective density matrix treatment.)
And, although it's well outside my area, I believe lots of people are trying things different to quantum mechanics. Just... String theory and others haven't (as far as I know!) come up with anything actually testable (or believable) yet. Quantum mechanics certainly isn't satisfactory (in that it leaves a lot to be desired in terms of understanding how things _really_ work, and it just can't work in strong gravitational fields when we start having to use General Relativity) - it just works so damn well (in that it predicts the outcome of experiments, behaviour of devices, and (so far) _everything_ we've been able to test).
Sorry I wasn't clear. Creating an exact copy of an unknown quantum state isn't possible; I believe the No-cloning-theorem - which, granted, is only a mathematical thing and is only true if quantum mechanics is - was referenced earlier. _If_ you know the state, you can make as many copies as you like. I did it this morning, in my lab.
>> Unfortunately, the best you can do is measure one of the atoms (or whatever) in >> the entangled pair and apply an operation to the other to make it exactly like the first.
>How is this not copying?
You destroy the state of the source in the process. This is akin to "mv source target", or at least "cp source target" then "rm source" as pointed out earlier in the thread.
I couldn't let this one pass. At no point do they create a copy of the original state. If you could do this, you could create yet another copy until you had as many as was necessary to measure whatever property you liked as accurately as you liked (even `conjugate' ones like the famous position and momentum or spin along different directions).
Unfortunately, the best you can do is measure one of the atoms (or whatever) in the entangled pair and apply an operation to the other to make it exactly like the first. The clever bit is that these entagled pairs could, in principle, be moved around classically, stored, and then used as a resource to do some teleporting later. i.e. You can't beam Kirk down to the surface unless you've already been there in the shuttle and left a machine full of some entangled stuff, the other half of which is still on the Enterprise.
Please don't just dismiss this. Cleverer people than you or me have been fighting with this for years. The uncertainty principle may be incorrect, but this experiment certainly does not suggest it.
This should confuse a few people. Group velocity (the apparent speed of a collection of waves - aka wave packet) can go faster than the phase velocity (the speed of a given wave making up a wave-packet).
\begin{rant}
Then they start talking about amplification in optic fibres, but the zero of intensity at the start of a pulse can't go faster than 299792458m/s so it can't carry information (and other such misleading things)...
\end{rant}
An image is displayed on an LCD screen by varying the darkness of each pixel, NOT by varying how much light each pixel emitts (as with a CRT). An LCD screen still needs a (very) bright white back-light.
An LCD screen displaying white will only be letting through about half the light from this back-light
A more efficient back-light has got to be good news for laptop battery life.
But what I wanna know is, does this mean we are looking away from the center of the universe?
Not as such. To picture the expansion of the univsere, think of all the galaxies, stars etc as small dots on the surface of a baloon. As the balloon is inflated, the area of it's surface, and the separation of the dots, expands. You can rotate the balloon so that you're looking at any dot you choose, and everything looks the same - there is no real centre to the 2 dimensional surface of the balloon. The only sensible definition of a centre is at point in 3D space where the expansion of the balloon started.
Similarly, there is no point in 3D space in our universe that could be considered it's centre; the only true centre of the universe must be the position in 4D space-time in the past, from which the expansion started. i.e. the big-bang is the centre of the universe.
Is there some crazy ball of energy still expanding outward or something?
Yes, but we can only see so far back as the universe was opaque very early in it's history; we can see the remnants of the big bang, but not the fireball itself.
Nope. In the very early Universe, all the matter was so hot that it was completely ionised. That is, there were lots of protons flying about and lots of electrons, just doing there own thing. It turns out, that light interacts very strongly with free electrons, so any light that was around at this early stage (such as from the big bang...) would've bounced around so much that it no longer carried any useful information about earlier times. Kind of like trying to see what the moon looks like through a really dense cloud.
Incidently, once the Universe cooled enough, light was able to pass through it. The light that started at this time is the oldest in the Universe and is what we now see as the Cosmic Microwave Backgound - far from being useless, this tells us huge amounts about the early Universe.
NASA's WMAP Mission site has a very good explanation.
Blackholes, rather than providing an area strong enough for particles to survive, seem to do the opposite. They're so small, and dense, that the gravitational field varies very quickly as you move away from it.
This means that, if you were near to the centre of the hole, your feet would be pulled on much harder than your head. If you were close enough, this would rip you in two. Move even closer, and it could rip apart the atoms that made you up.
I had the good fortune to be taught by Prof. Isham in the Foundations of Quantum Mechanics. He is a brilliant lecturer and, while his book cannot rival the lucidity of his lectures, it comes a close second.
Slashdot Mirror
I was going to reply point by point, but I really must do something else with my Sunday.
"I am not aware of any relativistic QM theory. In fact, I believe that to be one of the things most actively sought in modern physics."
General Relativity is the problem. Resolving Quantum Mechanics and Special Relativity (i.e. everything except gravity) has been done. It's called Quantum Field Theory. (Specifically Quantum Electrodynamics, or QED, for all things relating to the Coloumb force.) A lot has been done since the 1920s. I'd be surprised if you could get any useful molecular results from the Schrodinger equation.
`Particle in a box` is a toy problem, designed to introduce the student to the idea of boundary conditions. Imagine a guitar string, fixed at both ends, and that's your physical model. The `wave` isn't electromagnetic, gravitational or the like; it is a description of probability amplitudes, the square of which corresponds to probability (in the generally used interpretation).
The particle doesn't bounce off the walls. It is spread out, and doesn't rattle around like a classical ball. (If it did, you could measure the recoil as it hit the walls!)
As for quantisation emerging: the postulates do not include discrete eigenvalues. Read them again. Preferably from a better text book. It only emerges when you apply quantum mechanics to a a specific situation.
"...2+2=5..." The maths of QM isn't as hard as for number theory. And is certainly a lot more interesting.
"Physics courses in college take the same path." [They state unjustified assumptions.]
Have you taken one? Depending on the university, students are introduced the the subject in a bit of a rush, and only later (often in an optional course) come back to carefully examine the postulates.
Sure, assumptions are necessary to simulate a real situation (like a molecule). Maybe it's not a clean theory. Or maybe we just don't know how to frame the maths to do it efficiently on a computer. Eitehr way, just because we have to simplfy it for a specific situation does not render the whole approach invalid.
I stand by my comment that you know too little (you don't need years of study) to say it's wrong. As far as I can tell, it's not classical, so it doen't make sense/feel right, so you conclude it must be wrong. A lot of work has gone in to trying to reporoduce results with classical mechanics only, and we'd all embrace it, if only it would work.
In no particular order...
... also a clearly quantized entity."; "time-independent Shrodinger equation defines energy levels to be discrete eigenvalues"
Spin
====
Agreed - Spin cannot be derived from the Schrodinger equation alone, but it emerges naturally in the relativistic theory; there are far fewer ad hoc additions to QM than you suggest.
The Schrodinger equation is incompatible with relativity. Schrodinger knew this, but was unable to fix it; we needed Paul Dirac for that. For understanding the fundamentals of QM, and trying seriously to punch holes in it, a cursory understanding of non-relativistic QM is woefully inadequate.
Quantisation
============
Quantisation emerges - it isn't postulated. Eigenvalues can be continuous.
Take the momentum operator (which, as it's normally defined, just uses a differential to extract the momentum from the equation describing a travelling wave psi = e^(i (k x - omega t)). This wave describes a particle moving in free space. Nowhere have we suggested k should take only certain values and, hence, for a free particle, momentum k is continuous. Only when boundary conditions (i.e. particle in a box, or in the field around an atomic nucleus) are imposed do things start to become discrete. That quantities are discrete _emerges_ from QM, and is not deliberately built in to it.
"The hamiltonian is
Not so. For the travelling wave psi = e^(i (k x - omega t)), in free space (i.e. V(r) = 0), H psi = (hbar k)^2 / (2 m) - there is no mention of (hbar k) being constrained to take one of a certain set of values - it is continuous. If you impose, say, V(r) = 0 for (|r| a/2), you have a `particle in a box` scenario. The Schrodinger equation has to be zero at r = (+/-)a/2 (this is from the maths, not an additional postulate) and you find there are only certain waves which satisfy this condition. THEN momentum (and hence energy) becomes quantised.
Uncertainty Principle
=====================
As for the uncertainty principle... if things are decribed by waves, it is inescapable. See the Bandwidth Theorem. For the case of a wave, this just says that the shorter a wave in time, the less well known it's frequency. Imagine playing a 1kHz note for less than 1ms; you would be hard pressed to work out the frequency if you just had this short sample to work with.
This was part of the inspiration/justification for the uncertainty principle, and is certainly not the whole story.
Schrodinger Equation
====================
To clarify, the Schrodinger equation is H psi = i hbar (d/dt) psi - nothing more.
The Hamiltonian, H, can take many forms, but when it is first introduced it just uses the Classical Mechanics Hamiltonian and replaces the usual variabes (such as momentum) with the equivalent operators. More generally, it can be used to write down the total energy of two systems (such as an atom and a field) AND the interaction between them. This interaction term is used to describe emission/absorption of light by an atom, for example.
Bohr
====
As I stated, Bohr's approach is outdated; he had to make up many things to make a working model, but his model is relevant now only as a teaching aid. Incidently, it was the accellerating charges issue that made him propose there were certain `stable` orbits; something (like an electron around a nucleus or a planet around a star) in orbit is always accellerating towards the centre (i.e. nucleus or star) and hence, if it's charged (as an electron is), it should emit.
Slits
=====
Double slits - try searching for this on Google Scholar or the arXiv.
We physcists do like to keep things as simple as possible, and experiments have been done with apparatus very closely resembling Young's double slits, and not just in crystalography. Be careful when you talk of a conspiracy; all published scientific work is open to review, and the experiments are repeated independently. Such accusations could mark
I haven't read your comments carefully yet (I intend to, but it's nearly the weekend!). My comments may be less than coherent (see earlier comment about weekend) but, for now...
p he
Interesting point about matter being the thing that's quantised, and not the light. Pondering...
You may want to look at the derivation of the black body spectrum as evidence of photons. It _seems_ impossible to reproduce observations without quantising the light. (i.e. if you don't, the maths implies something like a hot coal radiates huge amounts of ultraviolet light, which it clearly doesn't) http://en.wikipedia.org/wiki/Ultraviolet_catastro
(I suppose this could be seen as a quantum effect in matter, not light, but if this is how light is made, it presumably suggests light is also quantised.)
Good point about energy & time -> amplitude & phase; I'll ponder that. Problem, as far as I recall, is that phase isn't easy to extract from a wavefunction (in the way that momentum, position etc can be extracted).
The double slit experiment works just as well with single electrons, and bigger things such as Carbon-60 molecules.
Quantum Mechanics does explain the orbitals of electrons in atoms. Very well, in fact. Bohr's first stab at a model has been outdated, and orbitals are generally calculated using the Schrodinger equation (where THAT comes from is another matter!) and the electrostatic potential of the nucleus. Things like spin and such are included, and the results agree with experiment. There's no `artificial' constraint on the types of orbitals allowed - they're just the solution to the equation in that potential.
The Bell inequalities really are worth the trouble. They provide some very compelling results contrary to hidden-variables and approachs with suggest the only real probability is our lack of knowledge. I'll refresh my memory of them if I get a chance. Unless something has been overlooked, probability really is a state of nature. I don't like it, but I can't find a way around it.
There are lots of alternative theories to QM, usually proposed by people with no formal training in the field. I don't know your background, but you'd better know your conventional QM if you want to convince someone else to try to fill in the gaps in your theory. There are many people with the ability to either shoot this down, or make it in to something complete. (I am not one of these people!) i.e. I know lots who would be capable, but it mightn't be easy to convince them to spend their time trying.
Some of Feynmann's interpretations seem odd, but his Lectures in Physics series is excellent. The conceptual issues part of Chris Isham's work http://www.worldscibooks.com/physics/p001.html is also excellent (I had the good fortune to hear him lecture, and I hope it comes across half as well when written down). He deals with all the maths first, but the Conceptual Issues chapter can stand alone.
Hi again,
First off let me qualify this response by saying I'm answering in the context of quantum mechanics. As we both agree, this certainly isn't a complete theory of how the universe works. However, there appear to be a few misconceptions which I hope to help clear up so that they don't cloud the real issue of the inadequacies of QM...
>> You can't make a perfect copy of an individual quantum (i.e. one photon). If you
>> have an lot of quanta (i.e. loads of identical photons coming out of a laser)
>Those photons would be perfect copies of one another, wouldn't they? Ok, so the laser can't produce identical polarization, but it certainly can produce >identical phase and amplitude; that's what a laser beam is all about. Splitting a beam does not disturb those parameters and they can be separately >measured, as was done in the experiment. A measurement of the left beam would produce the same result as the measurement of the right beam, so you can >measure in one and know in the other. Phase and amplitude of a wave may be considered its equivalents of position and momentum, so doing this disproves the >uncertainty principle by example.
When talking about quantum mechanics involving photons, we usually consider their polarisation. This is because it is a variable which has only two components; it can be represented in a two-state basis[1]. When we make a measurement, we measure along one of these components. In the case of the beamsplitter I mentioned previously, we `split' the photon by sending the vertical component one way, and the horizonal component the other.
It isn't so neat for amplitude and phase. Firstly, they are not conjugate variables (like position and momentum), so measurement of one does not imply the other is disturbed. But let me come back to this...
Light polarised, say, linearly along a 45 degree axis to the vertical would have one component reflected and the other transmitted by the beamsplitter. This does suggest the amplitude of both outputs is equal (and so would the phase). If we use a mirror to shine these two outputs on to a screen, we will see interference. This result is consistent with waves, classical electromagnetism, common sense... there's no mention of quanta.
Unfortunately this does not correspond to duplicating the photon. Consider the photoelectric effect. http://en.wikipedia.org/wiki/Photoelectric_effect This is a good rugged experimental demonstration that light interacts in the discret lumps we call photons; the implication[2] is that it also only exists in these lumps. So, if we send in just one photon in to our beam splitter, which output does it come out of? It can't come out of both, but we see interference when we recombine them (and we don't if we block one of the paths).
What I've very awkwardly described is Young's Double Slit experiment. http://en.wikipedia.org/wiki/Double-slit_experimen t You probably know of this, and it's no mystery with waves; but it works with individual quanta too. (i.e. only one quantum in the aparatus at any one time, but interference is still seen!) So... which slit does the quanta go through?
If you're able to resolve the double slit experiment and the photoelectric effect without invoking superpositions, I'd be very interested to hear.[3]
[1] A basis is a set of states which we use to represent the value of some variable. For polarisation, there are several including: linear vertical & linear horizontal (i.e. the electric field of the light points up and down or left and right) and left circular & right circular (i.e. the electric field traces out a helix with one of the two possible helicities). These states are orthogonal in the same way as the x- and y- axes on a grid are; no amount of x can change the position in y.
[2] Maybe this is the stumbling block, but I'm sur
Now I've had my coffee, let me add to my last comment...
You can't make a perfect copy of an individual quantum (i.e. one photon). If you have an lot of quanta (i.e. loads of identical photons coming out of a laser) you can make copies, because you can measure lots of quanta and take the average to work out (to an accuracy limited only by the number of quanta you have) all the parameters you need to set up your equipment to make copies. This is all because you destroy (or at the very least disturb) each quanta when you measure it.
In the case of a photon, you'd probably (in principle) use a polarisation-sensitive beam-splitter and two detectors, one on each output from the beam-splitter. The photon's polarisation isn't just disturbed - the photon is absorbed and hence destroyed when it hits the detector and is measured. If (as would be the general case) it was in some superposition of both states, we'd still only measure either one state or the other, with the appropriate probability; you'd need lots of states to work out what this probability was.
In a laser, the photons it emits are all identical (in principle), but their polarisation is set by the geometry of the device or by spontaneous emission of photons from within the gain medium of the laser. Unfortunately, it's not enough to set the laser up, send in one arbitrarily polarised photon, and expect many copies of it to be emitted.
This isn't just a technical problem. A device which amplifies some signal from an individual quantum necessarily introduces noise, as do all amplifiers. This most you can get out of this device at the end is a classical probability, rather than the coefficients of a quantum superposition (complete with phase as well as amplitude).
Such a quantum amplifier would be needed if we were to actually make a macroscopic superposition, such as Schrodinger's famous cat in a box. The result of a radioactive isotope decaying or not, with 50% probability, would have to be amplified so that the result could be used to decide whether or not to break the beaker full of poison gas to kill the cat. Through the process of amplification, the quantum superposition is changed to a plain old classical probability, and the cat is either dead OR alive before we open the box - no observer required! (This tending towards a classical probability from a quantum one has a sound mathematical backing in the ugly but effective density matrix treatment.)
And, although it's well outside my area, I believe lots of people are trying things different to quantum mechanics. Just... String theory and others haven't (as far as I know!) come up with anything actually testable (or believable) yet. Quantum mechanics certainly isn't satisfactory (in that it leaves a lot to be desired in terms of understanding how things _really_ work, and it just can't work in strong gravitational fields when we start having to use General Relativity) - it just works so damn well (in that it predicts the outcome of experiments, behaviour of devices, and (so far) _everything_ we've been able to test).
Sorry I wasn't clear. Creating an exact copy of an unknown quantum state isn't possible; I believe the No-cloning-theorem - which, granted, is only a mathematical thing and is only true if quantum mechanics is - was referenced earlier. _If_ you know the state, you can make as many copies as you like. I did it this morning, in my lab.
>> Unfortunately, the best you can do is measure one of the atoms (or whatever) in
>> the entangled pair and apply an operation to the other to make it exactly like the first.
>How is this not copying?
You destroy the state of the source in the process. This is akin to "mv source target", or at least "cp source target" then "rm source" as pointed out earlier in the thread.
I couldn't let this one pass. At no point do they create a copy of the original state. If you could do this, you could create yet another copy until you had as many as was necessary to measure whatever property you liked as accurately as you liked (even `conjugate' ones like the famous position and momentum or spin along different directions). Unfortunately, the best you can do is measure one of the atoms (or whatever) in the entangled pair and apply an operation to the other to make it exactly like the first. The clever bit is that these entagled pairs could, in principle, be moved around classically, stored, and then used as a resource to do some teleporting later. i.e. You can't beam Kirk down to the surface unless you've already been there in the shuttle and left a machine full of some entangled stuff, the other half of which is still on the Enterprise. Please don't just dismiss this. Cleverer people than you or me have been fighting with this for years. The uncertainty principle may be incorrect, but this experiment certainly does not suggest it.
This should confuse a few people. Group velocity (the apparent speed of a collection of waves - aka wave packet) can go faster than the phase velocity (the speed of a given wave making up a wave-packet).
Physicsweb
American Institute of Physics
Sure as hell confuses me.
\begin{rant}
Then they start talking about amplification in optic fibres, but the zero of intensity at the start of a pulse can't go faster than 299792458m/s so it can't carry information (and other such misleading things)...
\end{rant}
New Scientist magazine ran an article examining the rather more than 5 senses we all have. I think, at last count, there were about 20...
Senses special: Doors of perception
Multimap here in the UK does a good job of combining maps with aerial photographs. It even overlays a transparancy of the map.
The photos are at least a few years old, but it's still pretty, if not particularly useful.
An image is displayed on an LCD screen by varying the darkness of each pixel, NOT by varying how much light each pixel emitts (as with a CRT). An LCD screen still needs a (very) bright white back-light. An LCD screen displaying white will only be letting through about half the light from this back-light A more efficient back-light has got to be good news for laptop battery life.
But what I wanna know is, does this mean we are looking away from the center of the universe?
Not as such. To picture the expansion of the univsere, think of all the galaxies, stars etc as small dots on the surface of a baloon. As the balloon is inflated, the area of it's surface, and the separation of the dots, expands. You can rotate the balloon so that you're looking at any dot you choose, and everything looks the same - there is no real centre to the 2 dimensional surface of the balloon. The only sensible definition of a centre is at point in 3D space where the expansion of the balloon started.
Similarly, there is no point in 3D space in our universe that could be considered it's centre; the only true centre of the universe must be the position in 4D space-time in the past, from which the expansion started. i.e. the big-bang is the centre of the universe.
Is there some crazy ball of energy still expanding outward or something?
Yes, but we can only see so far back as the universe was opaque very early in it's history; we can see the remnants of the big bang, but not the fireball itself.
... that we'll eventually see the big bang?
Nope. In the very early Universe, all the matter was so hot that it was completely ionised. That is, there were lots of protons flying about and lots of electrons, just doing there own thing. It turns out, that light interacts very strongly with free electrons, so any light that was around at this early stage (such as from the big bang...) would've bounced around so much that it no longer carried any useful information about earlier times. Kind of like trying to see what the moon looks like through a really dense cloud.
Incidently, once the Universe cooled enough, light was able to pass through it. The light that started at this time is the oldest in the Universe and is what we now see as the Cosmic Microwave Backgound - far from being useless, this tells us huge amounts about the early Universe.
NASA's WMAP Mission site has a very good explanation.
Blackholes, rather than providing an area strong enough for particles to survive, seem to do the opposite. They're so small, and dense, that the gravitational field varies very quickly as you move away from it.
This means that, if you were near to the centre of the hole, your feet would be pulled on much harder than your head. If you were close enough, this would rip you in two. Move even closer, and it could rip apart the atoms that made you up.
I had the good fortune to be taught by Prof. Isham in the Foundations of Quantum Mechanics. He is a brilliant lecturer and, while his book cannot rival the lucidity of his lectures, it comes a close second.