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Natural Science Forum / Physics / General Physics / July 2007Understanding Schrodinger's cat.
The central theme behind the cat paradox is that before we look in the box the cat is both alive and dead. See John Gribbin "In search of schrodinger's cat" page 203. In the original experiment we use a poison, in a more human version we use a sleeping aid (laughing gas) In that case the central theme becomes: before we look in the box the cat is both sleeping and not sleeping. Suppose we change the experiment the box is completely empty (no radio active decay) except we place a dish with milk in the box. We perform the experiment a thousand times. What will be the outcome ? In all cases the cat is not sleeping (alive) We perform the experiment again a thousand times but now with sleeping "power" in the milk. What will be the outcome ? In all cases the cat is sleeping (dead) (This explains why I prefer a more human version) Does it makes sense to claim that before we look inside the box that the cat is both sleeping and not sleeping (alive) I have my doubts. Now suppose we perform the original experiment with radio active decay but not with a poison but with laughing gas a thousand times and we look inside the box each time after 10 minutes. What will be the outcome ? 500 times the cat is not sleeping and 500 times the cat is sleeping. Suppose we look after 20 minutes. What will be the outcome ? 250 times the cat is not sleeping and 750 times the cat is sleeping. Does it makes sense to claim that before we look inside the box that the cat is both sleeping (dead) and not sleeping.(alive) Again I have my doubts. What this test tells us is something about the half-life of the radio active element used. No more no less. Does this test tells us something about the concept of superposition ? Before you want to answer that question you first have to explain what superposition is. IMO the test does not require the concept of super position as such it tells us nothing about superposition. Ofcourse I can be wrong. Nicolaas Vroom http://users.pandora.be/nicvroom/ > The central theme behind the cat paradox is that before we look > in the box the cat is both alive and dead. The central theme is that before we look in the box we don't know whether the cat is alive or dead and we give both states a probability of 50%. Dirk Vdm > > The central theme behind the cat paradox is that before we look > > in the box the cat is both alive and dead. > > The central theme is that before we look in the box we don't know > whether the cat is alive or dead and we give both states a probability > of 50%. The OP sems to be trying to add additional random events on top of that. In other words you need to work out the probability of the partical decaying and/or the cat drinking the milk with sleeping stuff in. Personally I cant' see what he's driving at. On Jun 30, 8:32 am, "Dirk Van de moortel" <dirkvandemoor...@ThankS-NO- SperM.hotmail.com> wrote: > > The central theme behind the cat paradox is that before we look > > in the box the cat is both alive and dead. > > The central theme is that before we look in the box we don't know > whether the cat is alive or dead and we give both states a probability > of 50%. That doesn't quite seem to cover it: a classical system may certainly be in a state known only in probability, the difference is (to play along with the game), the classical state is supposed to be a definite if unknown state, while the quantum system exists in a superposition of states. Not playing the game, and releasing the objection I am choking back, the quantum state is also a perfectly definite state: what is indefinite is merely the outcome of an observation for which the state -- which happens to be expressed as a linear superposition in a particular basis, but is not thereby distinguished from all other states in the Hilbert space, which share this property in some basis -- is not an eigenstate. Schroedinger's cat is merely a thought experiment which, like some other thought experiments (Mach's bucket) has somehow been elevated to a kind of physical principle, no longer requiring thought. > On Jun 30, 8:32 am, "Dirk Van de moortel" <dirkvandemoor...@ThankS-NO- > SperM.hotmail.com> wrote: [quoted text clipped - 4 lines] >> whether the cat is alive or dead and we give both states a probability >> of 50%. I shouldn't have used the word "state" here. Better had been "we give both possibilities a probability of 50%" > That doesn't quite seem to cover it: a classical system may certainly > be in a state known only in probability, the difference is (to play > along with the game), the classical state is supposed to be a definite > if unknown state, while the quantum system exists in a superposition > of states. That is just words that go with the model. > Not playing the game, and releasing the objection I am choking back, > the quantum state is also a perfectly definite state: what is [quoted text clipped - 3 lines] > states in the Hilbert space, which share this property in some basis > -- is not an eigenstate. Again - just words that go with the chosen model. "Until we have looked we give both possibilities a probability of 50%" > Schroedinger's cat is merely a thought experiment which, like some > other thought experiments (Mach's bucket) has somehow been elevated to > a kind of physical principle, no longer requiring thought. Exactly. Dirk Vdm >> On Jun 30, 8:32 am, "Dirk Van de moortel" <dirkvandemoor...@ThankS-NO- >> SperM.hotmail.com> wrote: [quoted text clipped - 18 lines] > Again - just words that go with the chosen model. > "Until we have looked we give both possibilities a probability of 50%" Until we have looked we do not know what the state is of the cat. (assuming the box is not made from glas) To find out what the probability is that the cat is still alive (after a certain amount of time) you have to perform this experiment a thousand times. In the above mentioned book at page 203 we read: "The apparatus in the box is arranged so that the detector is switched on for just long enough so that there is a fifty fifty chance that one of the atoms in the radioactive material will decay and that the detector will record a particle." That means you must know the half-life time of the radio active material involved. Implying that if you make the switched on time longer there is a larger chance (probablity) that the detector will record a particle and the cat is dead. Next we read: "According to the strict Copenhagen interpretation, just as in the two-hole experiment there is an equal probability that the electron goes through either hole and the two overlapping posibilities produce a superposition of states so in this case the equal probabilities for radioactive decay and no radio active decay should produce a superposition of states." For me the biggest question is can you really compare the two cases. In the first case there are two states involved i.e the left hole being open or closed the right hole being open or closed. In the second case there is only one state involved ie. decayed yes or no. What are the overlapping posiblities when radio active material is involved ? To learn more about the two slit experiment with electrons is there any one who can answer the following questions. Suppose my electron generator produces 60 electrons in 10 minutes. (To test this I place my detectors in front of the the two slits and I count the the number of electrons over a period of 10 minutes. The result is 60) Next I place the detectors behind the two slits, BOTH slits are open. I run the experiment for 10 minutes. The result again is 60 hits. (I.e. the same count before as behind the two slits, with both slits open) Next I close the Right slit. Left slit is open Q1 what will be the count after 10 minutes. Next I open the Left slit and close the Right slit. Q2 what will be the count after 10 minutes. Nicolaas Vroom http://users.pandora.be/nicvroom/ > In the second case there is only one state involved > ie. decayed yes or no. As I undrstand it that's not the case. The particle is both decayed and not decayed at the same time. On Jul 4, 3:10 pm, "CWatters" <colin.watt...@turnersNOSPAMoak.plus.com> wrote: > > In the second case there is only one state involved > > ie. decayed yes or no. > > As I undrstand it that's not the case. The particle is both decayed and not > decayed at the same time. A number of observations: "A and not-A at the same time", in connection with quantum mechanics, is a maximally spooky way of presenting a fundamentally unspooky idea. One must understand a little of the structure of quantum mechanics, that the state is associated with a vector, which lives (of course) in a vector space; and one must understand a little about vectors, in particular the idea that a given vector can be expressed as a sum ("linear superposition") of other vectors in an indefinate number of ways. One must also understand that certain sets of vectors, called "bases", or we should say certain bases, have the property that each element (basis element, basis vector) is associated with a physical state which is supposed to return a definite result under some kind of measurement. (Audience doses peacefully at this point). Anyway, "A and not-A at the same time" expresses the idea that a particular vector (state vector, state, wave function) can in fact be expressed as a combination of some basis vector which returns "A" with certainty, under the appropriate measurment, and another basis vector which returns "not-A". So far, this is about as spooky as saying that "northwest" is "north" and "west" at the same time. One additional feature intrudes, whereby when we take a measurement corresponding to the basis in which "north" and "west" are eigenvalues (possible definate outcomes of a measurment associated with this particular basis), which we might call the "Manhattan measurement", "northwest" is not a possible outcome -- perhaps the crow, at first flying over the street grid northwest, is now dropped to the ground and forced to walk -- and finds that he may either walk north, or walk west, but not both simultaneously. Finally, our spooked out Manhattanite, having prepared a large number of crows in the "northwest" state, and seeing that as he wings them and they fall the to ground, must hobble off either north, or west, and well ... really weirds himself out. Was the crow flying north or west before he "measured" it? Did it only begin walking north or west after we "observed" it? Secondly, and more briefly, it's not clear if "decayed" and "not- decayed" label states in a legitimate basis: without this structure we are not even free to speak of the state as being "both at the same time" in the same convoluted way we spoke of the crows. >>> On Jun 30, 8:32 am, "Dirk Van de moortel" <dirkvandemoor...@ThankS-NO- >>> SperM.hotmail.com> wrote: [quoted text clipped - 5 lines] >> I shouldn't have used the word "state" here. Better had been >> "we give both possibilities a probability of 50%" >> Again - just words that go with the chosen model. >> "Until we have looked we give both possibilities a probability of 50%" [quoted text clipped - 17 lines] > there is a larger chance (probablity) that the detector > will record a particle and the cat is dead. In fact the whole experiment is much more complex. In order to find the half-life time (this is not the correct definition) you need a geiger counter a piece of radio active material and a stop watch. You wait when you detect the first decay (click geiger counter) and then you start the stop watch. You hear the second click and you wait for the third thick. You stop the stop watch and you divide the time by two. (Tis means three alpha particles are involved) This gives you a rough estimate of the half-life time. To get a better estimate you wait until the 1001 ste click and then you stop the stopwatch and you divide by 1000. When you use this half-life time definition with your schrodinger's cat experiment you have a 50% chance that after when you look the cat has a 50% chance that he is either alive or dead. But what does it mean for your schrodingers cat experiment ? Also with that experiment you need a geigercounter and a stop watch. First you place the cat in the box and a screen between the radio active material.and the container with the gas such that the alpha particle can not break the glas. You wait untill you hear the first click then you start the stop watch and you lift the screen. You wait until the stop watch has reached half-life time. That means you have reached the point with a 50% chance that a particle has decayed. You stop the stop watch and you drop the screen in order to terminate the experiment. If thereafter there is a release of an alhpa particle, the cat will not die. (The part done by you above can be automated) During the first experiment where you calculated the half-life time you can ask your self if there is any form of superposition involved. I doubt that. Ofcourse the experimentor can claim that he at the beginning is in a superposition of states of not-knowing/knowing when the first click comes and then in not-knowing/knowing when the second click comes etc, but such a discovery does not give me an eureka feeling. You can also raise the same question during the schrodinger cat experiment as described. But again I have doubts. Nicolaas Vroom http://users.pandora.be/nicvroom/ > > The central theme behind the cat paradox is that before we look > > in the box the cat is both alive and dead. > > The central theme is that before we look in the box we don't know > whether the cat is alive or dead and we give both states a probability > of 50%. Excuse me. You think this is a reasonable statement? Please explain to me how having ZERO knowledge one can somehow assign a 50% probability to "states". That is nuts. It's not a coin in the box. It's not just Schrodinger's cat that's chasing it's tail! >> > The central theme behind the cat paradox is that before we look >> > in the box the cat is both alive and dead. [quoted text clipped - 4 lines] > > Excuse me. You are excused. Dirk Vdm >You think this is a reasonable statement? Please explain to > me how having ZERO knowledge one can somehow assign a 50% probability > to "states". That is nuts. It's not a coin in the box. It's not just > Schrodinger's cat that's chasing it's tail! On Jun 30, 8:11 am, "Nicolaas Vroom" <nicolaas.vr...@pandora.be> wrote: > The central theme behind the cat paradox is that before we look > in the box the cat is both alive and dead. [quoted text clipped - 3 lines] > In that case the central theme becomes: > before we look in the box the cat is both sleeping and not sleeping. I think you missed the whole point of the thought experiment. Whether the cat lives or dies is not the main issue. It's how it happens and if that event is linked to something that is already known to be in superposition (such as an atom decaying or not), then it has supposedly similar consequences for the cat. > Suppose we change the experiment the box is completely empty > (no radio active decay) except we place a dish with milk in the box. > We perform the experiment a thousand times. > What will be the outcome ? > In all cases the cat is not sleeping (alive) Yes, but this is absolutely nothing to do with Schroedinger's cat. > We perform the experiment again a thousand times but now with > sleeping "power" in the milk. > What will be the outcome ? > In all cases the cat is sleeping (dead) Yes again, but still having nothing to do with Schroedinger's cat. > (This explains why I prefer a more human version) > Does it makes sense to claim that before we look inside the > box that the cat is both sleeping and not sleeping (alive) > I have my doubts. In Schroedinger's original thought experiment, it makes some sort of sense. Because one could argue that the outcome is completely dependent on a quantum system that is already known to be in superposition. You could then argue that the cat's outcome is in a similar simultaneous set of states. > Now suppose we perform the original experiment with radio active decay > but not with a poison but with laughing gas a thousand times and we look > inside the box each time after 10 minutes. > What will be the outcome ? Depends on what you mean. The cat is now in a superposition of what? Laughing or not laughing? Sleeping or not sleeping? > 500 times the cat is not sleeping and 500 times the cat is sleeping. > Suppose we look after 20 minutes. > What will be the outcome ? Most likely sleeping, considering the amount of time that has passed. But that misses Schroedinger's original point too. > 250 times the cat is not sleeping and 750 times the cat is sleeping. > Does it makes sense to claim that before we look inside the > box that the cat is both sleeping (dead) and not sleeping.(alive) > Again I have my doubts. > What this test tells us is something about the half-life of the radio > active element used. > No more no less. But what does it tell you? I'm not sure where you're going with this. > Does this test tells us something about the concept of superposition ? > Before you want to answer that question you first have to explain > what superposition is. > IMO the test does not require the concept of super position > as such it tells us nothing about superposition. Many people misunderstand what Schreodinger was actually trying to say. Schroedinger's original thought experiment put the cat's fate in the same simultaneously superposed states that the radioactive decay had had quantum mechanically. Schroedinger wasn't literally saying that the cat was in superposition. Just that one could seriously interpret it that way and that told him that there might be something wrong with the way we were looking at QM. He agreed with Einstein that QM as we understood it was incomplete and devised his thought experiment to show how silly it could get. Unfortunately, I think that so many people take it way too literally these days. > On Jun 30, 8:11 am, "Nicolaas Vroom" <nicolaas.vr...@pandora.be> > wrote: [quoted text clipped - 10 lines] > if that event is linked to something that is already known to be in > superposition (such as an atom decaying or not), Why do you call this effect a superposition ? Any way what we are discussing is a radioactive sample in which many atoms can decay in a certain period of time. That means the sample can decay many times. Of course you do not know which paricular atom will decay. > then it has > supposedly similar consequences for the cat. >> (This explains why I prefer a more human version) >> Does it makes sense to claim that before we look inside the [quoted text clipped - 5 lines] > dependent on a quantum system that is already known to be in > superposition. Again please explain what you mean. > You could then argue that the cat's outcome is in a > similar simultaneous set of states. >> Now suppose we perform the original experiment with radio active decay >> but not with a poison but with laughing gas a thousand times and we look >> inside the box each time after 10 minutes. >> What will be the outcome ? > > Depends on what you mean. I mean you perform this whole experiment a thousand times and you write down what the outcome is: dead or alive ie. 1 dead 2 alive 3 alive 4 dead 5 alive etc. 1000 dead. >> 500 times the cat is not sleeping and 500 times the cat is sleeping. i.e. in total 500 alive and 500 dead. >> Suppose we look after 20 minutes. >> What will be the outcome ? [quoted text clipped - 12 lines] > > But what does it tell you? I'm not sure where you're going with this. You learn that how longer you wait the higher the chance is that the cat is dead. That is all what you can learn. Nothing about superposition. You can also make the box from glas. The result will be the same. >> Does this test tells us something about the concept of superposition ? >> Before you want to answer that question you first have to explain [quoted text clipped - 6 lines] > the same simultaneously superposed states that the radioactive decay > had had quantum mechanically. Radio active decay is a concept described by half-life time. which is specific for the radio active sample (atom) involved. If the half-life time is 1 hour than if you place the cat in the box for 30 minutes you have a 50% chance that the cat comes out alive. There is no superposition of states involved (as far as I can see) > Schroedinger wasn't literally saying > that the cat was in superposition. Just that one could seriously [quoted text clipped - 3 lines] > experiment to show how silly it could get. Unfortunately, I think > that so many people take it way too literally these days. Nicolaas Vroom http://users.pandora.be/nicvroom/ On Jul 2, 12:20 pm, "Nicolaas Vroom" <nicolaas.vr...@pandora.be> wrote: > > wrote: > >> The central theme behind the cat paradox is that before we look [quoted text clipped - 11 lines] > > Why do you call this effect a superposition ? Because that's how it is defined. According to the Copenhagen interpretation, all possible solutions to the appropriate quantum wave equations are in full linear combination, or superposition, until an observation is made. Once an observation has been made, one particular state is realized. > Any way what we are discussing is a radioactive sample > in which many atoms can decay in a certain period of time. > That means the sample can decay many times. > Of course you do not know which paricular atom will decay. This has absolutely nothing to do with what Schroedinger was trying to say. His thought experiment was about the superposition of the states of decay vs nondecay. > > then it has > > supposedly similar consequences for the cat. [quoted text clipped - 9 lines] > > Again please explain what you mean. Well, if the outcome of the decay is based on observation, then so is the cat's fate. It's that simple. > > You could then argue that the cat's outcome is in a > > similar simultaneous set of states. [quoted text clipped - 32 lines] > That is all what you can learn. > Nothing about superposition. True, but again, it's not relevant to what Schroedinger was trying to say. And that was what you were asking about. > You can also make the box from glas. The result will be the same. Not according to the Copenhagen interpretation, which essentially says that things don't happen (in terms of a particular state being chosen out of superposition) until you observe them. So the wave function would just collapses sooner for a transparent box because you wouldn't have to open it to make an observation. > >> Does this test tells us something about the concept of superposition ? > >> Before you want to answer that question you first have to explain [quoted text clipped - 14 lines] > There is no superposition of states involved > (as far as I can see) But there is. Half life is only a statistical concept spread over many many atoms. Individual atoms will not have a half life. But they do have a wave function governing radioactive decay, which is nothing more than the emission of a particular particle. In classical physics, the atom either emits the particle or it doesn't. It's not so simple in QM, where the solution of the wave equation can have more than one component that remain in superposition of possible states until observed. > On Jul 2, 12:20 pm, "Nicolaas Vroom" <nicolaas.vr...@pandora.be> > wrote: >> Why do you call this effect a superposition ? > [quoted text clipped - 3 lines] > observation is made. Once an observation has been made, one > particular state is realized. Suppose I shuffle a card deck, I place them face down in front of you and I point my finger to one of them and I ask you is that card red or black Before YOU look is that card also in a superposition of states red vs black ? >> Any way what we are discussing is a radioactive sample >> in which many atoms can decay in a certain period of time. [quoted text clipped - 4 lines] > say. His thought experiment was about the superposition of the states > of decay vs nondecay. decay vs nondecay of what ? one specific atom or the whole sample which contains thousands of atoms which can decay. (and which will decay in due time) >> You learn that how longer you wait the higher the chance is that >> the cat is dead. [quoted text clipped - 9 lines] > that things don't happen (in terms of a particular state being chosen > out of superposition) until you observe them. Do you mean that according to the Copenhagen interpretation the result will not be the same ? i.e. statistically relevant different (500 dead versus 500 alive in a box of steel versus 499 dead versus 501 alive in a box of glas is the same) > So the wave function > would just collapses sooner for a transparent box because you wouldn't > have to open it to make an observation. When ? When the cat dies ? and what happens if you open the steel box and the cat is not dead ? >> >> Does this test tells us something about the concept of superposition ? >> >> Before you want to answer that question you first have to explain [quoted text clipped - 23 lines] > than one component that remain in superposition of possible states > until observed. Assuming that that is correct the question again is when. Suppose the alpha particle released, rings a bell is that not the moment ? Suppose there are two observers one sees the cat first and calls: the cat is dead what about the second observer ? Nicolaas Vroom http://users.pandora.be/nicvroom/ Schrodinger's cat is just plain crazy thinking. It is a mind experiment. It can only prove that the cat in the box has a 50% chance of being alive or dead. Schrodinger used a quantum act to trigger the event that would take place inside the box. (so what?) Until we open the box only the cat knows if it is alive or dead (hopefully) When Hawking's was told of Schrodingers cat,it was said he reached for his gun. As far as the observer is concerned he must make a measurement to weed out all random states of a quantum system. If Einstein was alive today he would still be laughing at "Schrodinger's cat mind experiment" I am bert > Schrodinger's cat is just plain crazy thinking. It is a mind > experiment. It can only prove that the cat in the box has a 50% chance [quoted text clipped - 6 lines] > today he would still be laughing at "Schrodinger's cat mind experiment" > I am bert Why laugh at something intended to be instructive? > Schrodinger's cat is just plain crazy thinking. It is a mind > experiment. It can only prove that the cat in the box has a 50% chance > of being alive or dead. Well it's a bit more than that. In theory the wave equation for the whole system including cat could contain both states. It's wrong to think of it as a trigger. Imagine the process of trying to write equations for the whole system starting with a simple equation for the state of the radioactive particle and ending with an impossibly complex equation for the state of every atom and electron of the cat. The element of uncertainty in the equation for the particle causes uncertainty in the equation for the atoms of the cat. CWaters Yes what you posted is what Schrodinger had in mind when setting up this experiment. Reality is that cat was to big an object. He should have used a virus. Reason for that is a single virus can work in the two slit experiment, bert > > On Jul 2, 12:20 pm, "Nicolaas Vroom" <nicolaas.vr...@pandora.be> > > wrote: [quoted text clipped - 12 lines] > Before YOU look is that card also in > a superposition of states red vs black ? No, but nobody ever said that it was. But then what wave equation predicts the outcome of card shuffling? This is entirely a problem in the realm of classical probability. Quantum probability is fundamentally different, being the outcome of the wave equation governing the system. The main difference relates to the the notion that cards are classical objects that the rules of probability can be based on. In QM, supposedly no such fundamental objects exist outside of solutions to the wave equations. > >> Any way what we are discussing is a radioactive sample > >> in which many atoms can decay in a certain period of time. [quoted text clipped - 9 lines] > or the whole sample which contains thousands of atoms which can decay. > (and which will decay in due time) Any specific atom. This is governed by quantum probability rules. > >> You learn that how longer you wait the higher the chance is that > >> the cat is dead. [quoted text clipped - 14 lines] > (500 dead versus 500 alive in a box of steel > versus 499 dead versus 501 alive in a box of glas is the same) Essentially, yes. The Copenhagen interpretation says that, in the quantum realm, something doesn't actually happen until it is observed. This is called collapse of the wave function and it's meaning is still being debated today. It's not something that comes out of the mathematical apparatus, but has to be added on to explain how, while the mathematical solution to the wave equation is in superposition, we only see one particular outcome when an observation is made. There are other interpretations, but they come off as being just as strange or even stranger than Copenhagen, such as the many worlds view, which says that all possible outcomes are realized but in different worlds. Welcome to the wierdness of QM. > > So the wave function > > would just collapses sooner for a transparent box because you wouldn't > > have to open it to make an observation. > > When ? When the observation is first made. > When the cat dies ? When does the cat die? How would you know without looking? > and what happens if you open the steel box and the cat is not dead ? It means that no single atom has yet decayed. A perfectly valid solution to the equation. > >> >> Does this test tells us something about the concept of superposition ? > >> >> Before you want to answer that question you first have to explain [quoted text clipped - 27 lines] > Suppose the alpha particle released, rings a bell is that not > the moment ? Well, if you've set it up to ring a bell, hearing the the bell would be the act of observation that breaks the superposition. > Suppose there are two observers > one sees the cat first and calls: the cat is dead > what about the second observer ? The cat is dead. The wave function collapses upon initial observation. You need to understand that this slippery aspect of the Copenhagen School is still being furiously debated. Not everyone agrees with it and it's mostly a philosophical add-on to QM. Nobody has, to my knowledge, ever devised an actual experiment in this area and it's doubtful that anyone ever would be able to. While actions on the wave functions produce physical observables, the wave function itself is not an observable. >> On Jul 2, 12:20 pm, "Nicolaas Vroom" <nicolaas.vr...@pandora.be> >> wrote: [quoted text clipped - 13 lines] > Before YOU look is that card also in > a superposition of states red vs black ? The whole idea behind this remark is to define the concept of superposition i.e. to define in what type of processes we can speak of superposition en when not. If there is always superposition in volved then what is the value of such a concept. For example IMO it does not make sense to define the cat in a superposition of state of both alive and dead before you look in the box if you know that in the experiment no poison (and no radio active decay) is involved. That means the cat is always alive when you open the box. The same if you put the cat in a box already filled with poison. That means the cat is always dead when you open the box. For the card deck example I have the same problem. What is the physical significance to define superposition related to the state of cards in a card deck. In Nature of 5 July 2007 there are three articles related to quantum physics. At page 24 we read: "According to quantum theory a card parefectly balanced on its edge will fall down in what is known as a superposition - the card really is in two places at once. If a gambler bets money on the quenn landing face up, the gambler's own state changes to become a superposition of two possible outcomes - winning or losing the bet." This is based on Everett's theory. " At page 23 we read:. "The theory can be summed up by saying that the schrodinger equation applies at all times; in other words, that the wavefunction of the Universe never collapses." I have great problems with all of this. Accordingly to Everett's theory before you open the box the cat is always in a superposition of states of both dead and alive even if there is no poison involved. What is the physical significance of all of this ? What wories me the most how can you use such a vaque concept to build a Quantum Computer. At page 24 the following pressing question is raised: if everett's theory is science or philosophy. IMO the question is more if it is science with a big S or a small s. Nicolaas Vroom http://users.pandora.be/nicvroom/ On Jul 14, 2:55 pm, "Nicolaas Vroom" <nicolaas.vr...@pandora.be> wrote: > >> On Jul 2, 12:20 pm, "Nicolaas Vroom" <nicolaas.vr...@pandora.be> > >> wrote: [quoted text clipped - 73 lines] > > - Show quoted text - Particles are always in a superposition state, even if it is a trivial one. For example, a given cat could be in a superpositon state c1 | dead_function> + c2 |alive_function>, where |dead_function> is an eigenfunction with eigenvalue "Dead", and |alive_function> is an eigenfunction with eigenvalue "Alive", and and c1^2 + c2^2 =1. Maybe we could call the relevant operator the Life operator. Both | dead_function> and }alive_function> are eigenfunctions of the Life operator, but the superpositon state c1|dead_function> + c2| alive_function> isn't. The probability of finding the cat "Alive" is c1^2, and the probability of finding the cat "Dead" is c2^2. One could set up the experiment in such a way that either c1 or c2 is exactly zero. That make the wavefunction of the cat somewhat trivially a superposition state. The importance of superposition is that the eigenfunctions of an operator form a "Complete set", which means that form a complete basis for ANY function. You can write any function as a linear combination of these eignefunctions. I hope that answers your question. I though his cat behaved as if it was half dead. :-> Burt Schrodinger's cat can fit in both quantum,and cosmic worlds QM makes this possible. In our universe the cat is dead. In our parrel universe the cat is alive. bert > On Jul 14, 2:55 pm, "Nicolaas Vroom" <nicolaas.vr...@pandora.be> > wrote: [quoted text clipped - 101 lines] > > I hope that answers your question. At page 24 in the above mentioned document we read: "So Everett's theory is testable and so far agrees with observation. But should you really believe it." Does this mean that the "you" is also in a superposition state of QM is true versus QM not true ? And what about the auther of the article ? At page 17 in the paragraph "Seeing is believing" we read: "This is where the debate at the two meetings this year etc Although both events have been organized by Everett enthusiasts they are unlikely to be one sided affairs, as many researchers still have great doubts about his ideas." Does this imply does that each researcher is also in a superposition state ? (may be each is entangled in a mixture of superpositions). What are the c1 and the c2 values of each ? IMO the s of science in QM is even becoming smaller: there is a lot of speculation involved. Nicolaas Vroom http://users.pandora.be/nicvroom/ On Jul 15, 5:37 am, "Nicolaas Vroom" <nicolaas.vr...@pandora.be> wrote: > > On Jul 14, 2:55 pm, "Nicolaas Vroom" <nicolaas.vr...@pandora.be> > > wrote: [quoted text clipped - 117 lines] > in a superposition state ? > (may be each is entangled in a mixture of superpositions). No, not really. Really large particles essentially obey classical (or relative) mechanics. This is called the correspondence principle. The predictions of QM approach those of classical mechanics as quantum numbers become very large. The Schroedinger's cat thing merely shows that QM seems very strange for a classical particle. > What are the c1 and the c2 values of each ? > > IMO the s of science in QM is even becoming smaller: > there is a lot of speculation involved. No, there is no speculation at all - just calculation, which always gives the right answer. Hard to top that! > Nicolaas Vroomhttp://users.pandora.be/nicvroom/- Hide quoted text - > > - Show quoted text - > On Jul 15, 5:37 am, "Nicolaas Vroom" <nicolaas.vr...@pandora.be> > wrote: [quoted text clipped - 16 lines] > predictions of QM approach those of classical mechanics as quantum > numbers become very large. In the mentioned article in Nature at page 24 we read: "The gambler's own state changes to become a superposition of two possible outcomes - winning or losing the bet." That means for Nature this is a Yes. IMO for whatever it is worth, this is science with a small s. > The Schroedinger's cat thing merely shows that QM seems very strange > for a classical particle. The Schroedinger's cat is a thought experiment. To its bare minimum the experiment is a demonstration of the half live of a radio active element. Basically you need a geiger counter, a stop watch and a radio active element. After 1000 counts you stop the watch and you divide the time by 1000 to get the half-life time. Next you wait for the next decay (alpha particle) and you start the stop watch and you reset the geiger counter. You wait for the half-life time and then you place a shield between the radio active element and the geiger counter such that the alpha particles can not reach the counter. If you now or any time later observe the geiger counter you have a 50% chance that the counter is zero (or not 0) There is nothing strange in this. You can repeat this whole experiment over a long range of time and then you will see that the half-life time is increasing. There is also nothing strange in that. >> What are the c1 and the c2 values of each ? >> [quoted text clipped - 3 lines] > No, there is no speculation at all - just calculation, which always > gives the right answer. Hard to top that! The only calculation in the above mentioned experiment is to divide the time on your stop watch by 1000 in order to get the half-life time. Science with a small s starts when you claim that the state of the geiger counter is in a superposition state of zero and non zero after you have shielded the geiger counter (without looking at the counter). IMO such a claim has no physical significance. To be more more specific: You can not build a Quantum Computer using that concept i.e. half-life time of a radio active element. Nicolaas Vroom http://users.pandora.be/nicvroom/ > The central theme behind the cat paradox is that before we look > in the box the cat is both alive and dead. Try Again http://scienceworld.wolfram.com/physics/SchroedingersCat.html http://en.wikipedia.org/wiki/Schr%C3%B6dinger's_cat >> The central theme behind the cat paradox is that before we look >> in the box the cat is both alive and dead. > > Try Again > http://scienceworld.wolfram.com/physics/SchroedingersCat.html > http://en.wikipedia.org/wiki/Schr%C3%B6dinger's_cat Both documents are not in agreement with "Igor" who wrote: "He (Schroedinger) agreed with Einstein that QM as we understood it was incomplete and devised his thought experiment to show how silly it could get." The "same" is written in the afore mentioned book at page 205: "Schroedinger thought up an example to establish that there is a flaw in the strict Copenhagen interpretation, since obviously the cat cannot be both alive and dead at the same time" On the other hand the next line puts this in doubt: ''But is this any more "obvious" than the "fact" that an electron cannot be both a particle and a wave at the same time?" Is there anyone who can explain what "a superposition of states means" and describe a good example to demonstrate this? IMO radio active decay has nothing to do with this. Nicolaas Vroom http://users.pandaora.be/nicvroom/ > Is there anyone who can explain what > "a superposition of states means" It means being in multiple states at the same time. > describe a good example to demonstrate this? I believe that would be the double slit experiment. Light goes through both slits at once. You get interference fringes even when the light consists of just one photon. The light can be said to be in a superposition of states - the left slit state and the right slit state :-). However if you try to observe which slit the photon went through the interference fringes vanish as the photon is forced into one state or the other. Can't mix microscopic and macroscopic levels. Cats behave differently than atoms. > Can't mix microscopic and macroscopic levels. Cats behave differently > than atoms. Funny thing. That's exactly what Schroedinger was trying to say. > > Can't mix microscopic and macroscopic levels. Cats behave differently > > than atoms. > > Funny thing. That's exactly what Schroedinger was trying to say. Then he and I agree.. Only he can't state his case too well anymore. > > Is there anyone who can explain what >> "a superposition of states means" [quoted text clipped - 12 lines] > observe which slit the photon went through the interference fringes vanish > as the photon is forced into one state or the other. There is an interesting difference between double slit experiment of single photons and the Schrodinger's cat experiment (See my discussion with Igor) In the first there is only an active experimentor involved There is no "active" observer involved. The experimentor opens and closes the slits starts the experiment and stops the experiment when for example 100 photons are emitted. The observer observes a "photo" which shows for example interference fringes (both slits open) when the experiment is finished. In the second experiment there is both an active experimentor and an active observer involved. The experimentor puts the cat in the box and starts the experiment. The observer opens the box (which stops the experiment) and looks inside. Accordingly to QM: before that moment the cat is in a superposition of states (dead and alive), there is a collapse of the wave function, the superposition vanishes and the cat is either dead or alive. It is not clear to me when there is a collapse of the wave function in the double slit experiment. Nicolaas Vroom http://users.pandora.be/nicvroom/ On Jul 3, 10:49 am, "Nicolaas Vroom" <nicolaas.vr...@pandora.be> wrote: > > > Is there anyone who can explain what > >> "a superposition of states means" [quoted text clipped - 39 lines] > It is not clear to me when there is a collapse of the > wave function in the double slit experiment. Sure there is. It all depends on whether you want to detect particles or waves at the slits. > There is an interesting difference between double slit experiment > of single photons and the Schrodinger's cat experiment [cut] > It is not clear to me when there is a collapse of the > wave function in the double slit experiment. Consider a single photon incident on the screen. Where is it detected? The wavefunction is the two-slit pattern, while the photon is detected at a single point. If one is to invoke "collapse of the wavefunction", this is when it happens, when the photon is detected. The points of the low-intensity (eg one photon at a time) double-slit experiment are to show: (a) Diffraction doesn't depend on interaction between photons (leading to "a photon can only interfere with itself") (b) You can experimentally determine the magnitude of the wavefunction by repeated measurements of identical systems. Whether the experimenter sees this in almost-real time via a CCD, or looks later at a developed photo doesn't matter.  SignatureTimo Nieminen - Home page: http://www.physics.uq.edu.au/people/nieminen/ E-prints: http://eprint.uq.edu.au/view/person/Nieminen,_Timo_A..html Shrine to Spirits: http://www.users.bigpond.com/timo_nieminen/spirits.html >> There is an interesting difference between double slit experiment >> of single photons and the Schrodinger's cat experiment [quoted text clipped - 12 lines] > (a) Diffraction doesn't depend on interaction between photons (leading to > "a photon can only interfere with itself") leading to if there is an interference patern with single photons: Each photon "goes through" both slits ? > (b) You can experimentally determine the magnitude of the wavefunction by > repeated measurements of identical systems. > > Whether the experimenter sees this in almost-real time via a CCD, or looks > later at a developed photo doesn't matter. Nicolaas Vroom http://users.pandora.be/nicvroom/ > "Timo A. Nieminen" <timo@physics.uq.edu.au> schreef: >> [quoted text clipped - 17 lines] > leading to if there is an interference patern with single photons: > Each photon "goes through" both slits ? There is the (old, non-relativistic) quantum mechanical picture of particle and wavefunction that tells you where the particle might be found if looked for. The "particle goes through both slits" view is very Copenhagen. Bohm (or perhaps I should say Bohmians) would say otherwise. A Schroedingerite would say otherwise - you don't know which slit it went through, rather than saying it went through both. The maths of all pictures only tells you the probability of detection at location X. From a QED picture, you have what are essentially classical fields, and exchange of energy etc between them and other entities (essentially, other classical fields) is quantised, and instant and point-like. Thus, an electron becomes not a particle whose position is predicted by a wavefunction, but a quantum of excitation/de-excitation of a classical wave mode. The question of which slit the electron/photon/whatever goes through is meaningless; it isn't a particle in any classical billiard-ball sense. Thinking of quantum "particles" as classical "particles" can very rapidly lead you astray. Hanbury Brown's "Boffin" has a nice bit on reaction to the Hanbury Brown-Twiss intensity interferometer work, form the QM community. See also Lamb's "Anti-photon" and the "photon" special section in a few-years-old Optics and Photonics News. SignatureTimo Nieminen - Home page: http://www.physics.uq.edu.au/people/nieminen/ E-prints: http://eprint.uq.edu.au/view/person/Nieminen,_Timo_A..html Shrine to Spirits: http://www.users.bigpond.com/timo_nieminen/spirits.html >> There is an interesting difference between double slit experiment >> of single photons and the Schrodinger's cat experiment [quoted text clipped - 6 lines] > single point. If one is to invoke "collapse of the wavefunction", this is > when it happens, when the photon is detected. Okay. That means if I perform the double slit experiment with a 100 single photons there are 100 "collapses of the wavefunction" each time when a single photon hits the screen (or a pixel of a CCD) and the photon ceases to exist. Implying no human involvement. Accordingly to that same reasoning in the Schrodinger Cat experiment why is there not only a collapse of the wavefunction when (after decay) the alpha particle hits the container and the poison is released ? (or a bell rings? or a geiger counter increases ?) Implying again no human involvement. > Timo Nieminen - Home page: http://www.physics.uq.edu.au/people/nieminen/ > E-prints: http://eprint.uq.edu.au/view/person/Nieminen,_Timo_A..html > Shrine to Spirits: http://www.users.bigpond.com/timo_nieminen/spirits.html Nicolaas Vroom http://users.pandora.be/nicvroom/ > >> There is an interesting difference between double slit experiment > >> of single photons and the Schrodinger's cat experiment [quoted text clipped - 27 lines] > > Nicolaas Vroomhttp://users.pandora.be/nicvroom/ Hi Nicolaas, Look at: http://www.decoherence.de/ Over the past 2 decades. Our understanding of the quantum has increased a hundred fold. We now know there is no collapse of the wavefunction. Instead there is decoherence. This means even macroscopic objects have quantum effects and so we are back to the problems of the Schroedinger Cats in full force because decoherence entails that there is no collapse.. meaning even macroscopic objects should be in superposition even in decohered states. Now the challenge is to understand how come macroscopic objects don't produce superpositions. It has something to do with the "Problem of Preferred Basis" or why in the endless possibilities in Hilbert Space, the classical world is the prefered basis. To illustrate. What decoherence shows us is that the Statue of Liberty should always change whenever each one of us look at it. But why is the image not changing. The mystery is what the current batch of quantum researchers are solving. Those who are not updated with quantum research in the past decade or two won't be aware of the incredible changes going on in the quantum theory field. Jules > > There is an interesting difference between double slit experiment > > of single photons and the Schrodinger's cat experiment [quoted text clipped - 12 lines] > (a) Diffraction doesn't depend on interaction between photons (leading > to "a photon can only interfere with itself") How do you feel about this claim? Might it not be better to say that the field, which from time to time manifests itself as photons, interferes only with itself? Is not the photon a quantized excitation of the (single) field? >> (a) Diffraction doesn't depend on interaction between photons (leading >> to "a photon can only interfere with itself") [quoted text clipped - 3 lines] > Might it not be better to say that the field, which from time to time > manifests itself as photons, interferes only with itself? It certainly might be better to say that. > Is not the > photon a quantized excitation of the (single) field? At least sometimes. How about a quantised excitation of a monochromatic component of the field? How about a quantised excitation of a monochromatic plane-wave component? But yes, there is one field, and it tells us the classical energy density and flux, and thus the probability of detection of photons. I wonder why people say that the classical EM field _cannot_ be a wavefunction for photons? I recently put a newish paper by Bialynicki-Birula on this stuff onto my to-read pile; haven't reached it yet. (Was in Acta Physica Polonica B, which is freely available online. I had a quick look to see which issue, but didn't find it. Some possibly interesting stuff in the May Brownian motion issue I should download and read. Either browse many months of tables of contents - which you might enjoy anyway - or look on Iwo Bialynicki-Birula's webpage, which probably has the paper available, and many others you might like. Interestingly, he did a paper on rotational frequency shift in 1997, so he was thinking about it at the same time that we were, despite being in a quite different field.) How do I feel about the claim? It's serviceable, for most purposes. It contains some truth, even - it's a better description than anything that says "photon A interferes with photon B". Whether it works for entangled states, non-classical interference etc, I don't know. Maybe for a suitable definition of "photon", allowing for a multi-photon photon ... SignatureTimo Nieminen - Home page: http://www.physics.uq.edu.au/people/nieminen/ E-prints: http://eprint.uq.edu.au/view/person/Nieminen,_Timo_A..html Shrine to Spirits: http://www.users.bigpond.com/timo_nieminen/spirits.html > >> (a) Diffraction doesn't depend on interaction between photons (leading > >> to "a photon can only interfere with itself") [quoted text clipped - 15 lines] > of detection of photons. I wonder why people say that the classical EM > field _cannot_ be a wavefunction for photons? I'm happy you wonder that, because I've asked that question in the past, and received some authoritative answer why the classical field is not the wave function. I'm kind of a QM kook, as you might have noticed, because I harbor the expectation that when the smoke clears a bit, many of the things some were in a rush to throw out, like "classical logic", were working just fine, and continue to work just fine, and there is a comprehensible picture. In this regard I have a twinge of feeling about Feynman similar to what many of our friends harbor about Einstein: an excellent technician, his penchant for wallowing in "nobody understands it" has done great damage to thought. Something else I'd like to throw out is the facile distinction between "quantum" and "classical", which so many enjoy casually dropping. In my view, there is simply physical theory, and we are better off trying to put all physical theories on the same board. It's not that goddamn hard, goddamn it. > I recently put a newish > paper by Bialynicki-Birula on this stuff onto my to-read pile; haven't [quoted text clipped - 13 lines] > states, non-classical interference etc, I don't know. Maybe for a suitable > definition of "photon", allowing for a multi-photon photon ... I re-noticed recently that "entangled" maps very closely to "non- seperable" distribution in probability theory, aka "correlated", but I haven't drawn all the lines. Confusingly, although the technical definition of "entangled" seems to be exactly that for "correlated", the result somehow goes beyond the correlation we know: entangled states produce a correlation structure for observables incompatible with any given multi-variate distribution: I don't understand this mechanically. Perhaps shades can continue to read papers -- then I'll have an infinite amount of time. >>>> (a) Diffraction doesn't depend on interaction between photons (leading >>>> to "a photon can only interfere with itself") [quoted text clipped - 19 lines] > past, and received some authoritative answer why the classical field > is not the wave function. Do you recall the authoritative answer? Generally avoiding quantum stuff, I only look at such things out of interest, not out of pressure to generate papers, etc, so haven't devoted that much time to it. Of course, there's this big crossover between classical and quantum in my field, especially when it comes to angular momentum, so one can't avoid it altogether. > I'm kind of a QM kook, as you might have noticed, because I harbor the > expectation that when the smoke clears a bit, many of the things some [quoted text clipped - 6 lines] > technician, his penchant for wallowing in "nobody understands it" has > done great damage to thought. I never had the impression that either wallowed in NUI. Relativity-NUI is more an Eddington-thing, iirc. Feynman, well, read more Feynman. Feynman's general writing is IMHO a very good source on the nature of science, research, and all that. "Shut-up-and-calculate" does not sum up his philosophy. A funny thing is that a lot of the best science-side contributors to the philosphy of science wrote many anti-philosophy statements. > Something else I'd like to throw out is the facile distinction between > "quantum" and "classical", which so many enjoy casually dropping. In > my view, there is simply physical theory, and we are better off trying > to put all physical theories on the same board. It's not that goddamn > hard, goddamn it. Well, there is a major disconnect between classical and quantum theories. "Classical" is an imprecise description, since there are two main types of classical theories: classical mechanics and classical field theories. Classical mechanics is the short-wavelength version of quantum theory, and classical field theories are the high-quantum-number version of quantum field theories. In many ways, I think the two main branches of classical theory are less closely related to each other than they are to their respective quantum theories. >> How do I feel about the claim? It's serviceable, for most purposes. It >> contains some truth, even - it's a better description than anything that [quoted text clipped - 10 lines] > with any given multi-variate distribution: I don't understand this > mechanically. I keep asking people why "non-classical" interference is non-classical, but haven't gotten any particularly useful coherent answer yet (any lurkers who wish to give one?). Fundamentally, photons are _not_ little microscopic billiard balls, so perhaps there's a laguage disconnect, depending on whether the speaker is coming from a classical particle background, or a classical field theory background. Do seek out Hanbury Brown's book "Boffin", where he writes about the furore that the Hanbury Brown-Twiss interferometer stuff kicked up in the quantum optics community. A good book, even without that. On the topic of worthwhile reading, R. V. Jones's book, title might be "War most secret", is also excellent. > Perhaps shades can continue to read papers -- then I'll have an > infinite amount of time. :) A cute thing about the "a photon only interferes with itself" is that the experiments demonstrating that two independent sources interfering with each other (RF, optical, both have been done) utterly kill the "billiard-ball" photon - a billiard ball can't come from two sources at once. The idea of quanta as quanta of excitation of classical fields has no problem, since there is only one classical field, to which both sources contribute. SignatureTimo Nieminen - Home page: http://www.physics.uq.edu.au/people/nieminen/ E-prints: http://eprint.uq.edu.au/view/person/Nieminen,_Timo_A..html Shrine to Spirits: http://www.users.bigpond.com/timo_nieminen/spirits.html > > I'm happy you wonder that, because I've asked that question in the > > past, and received some authoritative answer why the classical field > > is not the wave function. > > Do you recall the authoritative answer? Alas no; probably because I didn't understand it. Could it have been something simple-minded, like "the electron wave function in three space tells you the probability of finding a particular electron, the electromagnetic field tells you..." what? Clearly it has something to do with an indefinite number of photons, not a single photon (a single photon wavefunction?). Since, as Feynman says somewhere, photons and electrons are essentially identical in all respects as quantum particles (well, aside from mass, charge, and spin), there ought to be something analogous to a "many photon wave function", which would live in 3n, not 3-space. But then we have ventured into quantum field theory, where "n" is not even specified. In other words, I don't know. Maybe the electromagnetic field is a particular kind of summary of the "many photon wave function": a reduction in information. > Generally avoiding quantum stuff, > I only look at such things out of interest, not out of pressure to [quoted text clipped - 8 lines] > > fine, and continue to work just fine, and there is a comprehensible > > picture. In my view, the rush to question "logic" betrays a general, dare I say, philosophical confusion. It's not all that hard to lay out a general, shall we say "null philosophical theory" (well, not that easy either, but I think not that hard) which makes almost no assumptions about what we are accomplishing by imposing science on our sensory world. We need not assume that things exist, or don't exist, or even give meaning to these statements. All we need look for is a logical (i.e. "mathematical") structure underlying our sensory impressions. This "model" is subject to some layered assumptions, among which is the very lowest is "sanity": i.e., the ability to reason (for some this assumption is faulty, but we must make it to procede). Playing with logic may not indicate a sterile questioning of sanity (if we can't reason, then we might as well give up an reach for the drool bucket), but is maybe more a cute mathematical game: however, label intermediates provocatively as we will ("both exists and doesn't exist at the same time"), in the end our structure must make predictive contact with observation. Sorry ... it is an old rant, and the impossibility of putting it into 100 words or less never deters me. ;-) > > In this regard I have a twinge of feeling about Feynman similar to > > what many of our friends harbor about Einstein: an excellent [quoted text clipped - 3 lines] > I never had the impression that either wallowed in NUI. Relativity-NUI is > more an Eddington-thing, iirc. Poorly worded on my part. I was not implying Einstein wallowed in NUI (good phrase), but merely drawing a parallel between my iconoclasty and that of the Einstein nut jobs. I see you don't agree with my characterization of Feynman either. I haven't read as deeply into his works as you, but when introductory lectures are larded with phrases like "nobody understands it!", one might be left with the impression of wallowing. My annoyance is coupled with his fame: when a man of his caliber says "nobody understands it!", beyond a frank admission of human frailty, one might form the idea that it is impossible to understand it at all, maybe even a fool's errand to try. This attitude -- whether or not I am fairly interpreting Feynman -- is lazy and defeatist: though maybe justified in the case of those who hold it. That's all the diconnected rant time allows me now; I shall return to the balance of your reply as rant-fodder later. :-) N V Superposition means two in the same place. it goes against Pauli's principle. Still QM tells us an electron can be in to different places at the same time,or two electrons can fit in the same spot. Thus in the weirdness of the quantum realm bert > N V Superposition means two in the same place. No, that's not what it means. "Superposition" means that the wavefunction for the particle is a linear combination of eigenfunctions, but the overall wavefunction might NOT be an eigenfunction. The probability that we will m easure a particular eigenvalue depends on the square of the coefficient of the corresponding eigenfunction in the linear combination. In other words, "superposition" is a mathematical concept that leads to interesting results as far as measurement goes, quantum mechanically. > it goes against Pauli's > principle. Still QM tells us an electron can be in to different places > at the same time,or two electrons can fit in the same spot. Thus in the > weirdness of the quantum realm bert > N V Superposition means two in the same place. it goes against Pauli's > principle. Still QM tells us an electron can be in to different places > at the same time,or two electrons can fit in the same spot. Thus in the > weirdness of the quantum realm bert Rather than looking like a complete imbecile on usenet, it's usually wise to look things up and get things right before you actually post. |
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