The Knowable and The Unknowable in quantum mechanics

Discussion in 'Physics & Math' started by lethe, Nov 12, 2002.

  1. lethe Registered Senior Member

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    When heisenberg proposed his interpretation of quantum mechanics, it was met with some skepticism in the scientific community. the most outspoken critic was einstein. perhaps you have heard his quote "God does not play dice". specifically, he is asserting his dissent from the orthodox view of quantum mechanics, which claims that nature is inherently probabilistic, and small particles do not exist in particular states, but in many states at once, each state having a probability of being measured.

    In the early days, einstein used to devise elaborate gedanken experiments wherein it was possible to know a particles momentum and position exactly. the quantum physicists would have to show him how his experments would fail, and in general, it was very difficult. einstein was very elaborate in the construction of apparatus that could be used for measurement. eventually, all his arguments were defeated. the culmination was when bohr used einsteins own theory of general relativity to show that einsteins measurements changed the state of the particle just enough to satisfy the heisenberg uncertainty principle.

    at this point, einstein et al. adopted a weaker position. there could no longer be any doubt that quantum mechanics (and the uncertainty principle) were correct. but they believed that quantum mechanics was incomplete. that there was more information in the system, that was for some reason hidden.

    this position is sometimes known as "realism". the position described above is sometimes known as "the orthodox position". einstein, schrödinger, bohm, et al. were realists. included in their number were many of the early fathers of quantum mechanics.

    let me quote schrödinger in his gedanken experiment to show the absurdity of the orthodox position.

    this is known as schrödinger s paradox. to summarise, we put a cat in a box, and tie his fate to a quantum event, which is governed by quantum indeterminacy. if we perform no measurements on the system, then the cat must be simultaneously dead and alive. schrödinger (and others) regarded this as patent nonsense. the cat must either be dead or alive, it doesn t check to see whether anyone is looking before it decides whether to live or die.
     
    Last edited: Nov 23, 2002
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  3. lethe Registered Senior Member

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    schrödinger s paradox is of some concern. i will return to it.

    let me now move on to the EPR paradox. einstein accepted the heisenberg uncertainty principle as correct, but devised this gedanken experiment to show that quantum mechanics must be incomplete; that there must be more information in the system than what we can measure.

    consider the decay of a pion into two electrons. in order to conserve angular momentum, the spins of the two electrons must be anti-aligned. so if we measure that one electron has spin up, then any measurements on the other electron must yield spin down.

    under the orthdox position, we would say that both electrons were in both the spin up and spin down states, and we measured spin up because it had a 50/50 chance of being measured. it was not necessarily in the state we measured.

    so let the electrons fly apart, until they are one light year apart. the are both in a combination of states. we measure one of the electrons spins, and discover it is spin up. this measurement changes the wavefunction of the two electrons, changing the one we measured into a spin up state. it changes the other electron to a spin down state, at the very same instant as we measured the first one. if it did not happen at exactly the same instant, then i could find a time when the conservation of angular momentum was violated.

    somehow the information of the first measurement was sent to the second electron instantaneously. however, relativity requires that no information can move faster than light. if you find some means of transfering information faster than light, then i can boost to a frame of reference where the information arrived before it was sent. you could conceivably get the answer to your message before you sent it. and then choose not to send it. this is a clear paradox.

    this is the EPR paradox, due to einstein, podolsky, and rosen.
     
    Last edited: Nov 23, 2002
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  5. Prosoothus Registered Senior Member

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    lethe,

    I just wanted to state that the Schrodinger's Cat Experiment isn't a unique situation that can't occure in everyday life. If uncertainty really exists in the subatomic world it would also exist in classical mechanic and macrophysics since everthing is built from subatomic particles.

    Life forms and other complex forms of matter are proof that uncertainty in the subatomic world does not exist.

    Tom
     
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  7. Prosoothus Registered Senior Member

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    lethe,

    I have an easier explanation. Let's say the pion is composed of six-dimensional subatomic particles. Let's also say that subatomic particles can only stack themselves in three dimensions. Therefore, three of the pions dimensions would be in our universe, while the other three would be in an alternate universe. Now if the pion decays into two electrons in our universe, that doesn't mean that it decayed into two particles in the other universe. So if the pion remains intact in the other universe, while it decayed into two electrons in this universe, each electron would know what the other was doing since they are both still the same particle in the parallel universe. In other words, the information sent from one electron to the other is sent interdimensionally. As a result, the speed of light rule is not broken.

    Now, as for relativity, it's wrong.


    Tom
     
  8. lethe Registered Senior Member

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    The EPR paradox was regarded as proof that the electrons must actually have spin up and spin down, seperately, to preserve causality. then the heisenberg uncertainty principle tells us only that we cannot measure different spins simultaneously. not that the particles don t have definite spin states. just that those spin states are unknowable.

    I will return to the EPR paradox.

    now it is on to bell s theorem. let us assume that the orthodox position is incorrect, and that einstein s realism is correct. then the electrons have some hidden information, that we cannot measure, but nevertheless exists.

    let us label this hidden information with a variable x (usually the variable is written \lamba, but for obvious reasons ...). this is the hidden variable.

    Now consider a modified version of the EPR setup. instead of measuring the spins of the two seperate electrons along some common axis, let us choose independent directions to measure the spins along for the two electrons. call these directions u and v (unit vectors). then let P(u,v) be the average of the product of the spins. if u and v are parallel, then i recover the original EPR experiment. conservation of angular momentum requires that the spins be anti-aligned, so the product of the spins will always be -\hbar^{2}/4. or in units of \hbar/2, P(u,u)=-1. it is easy to show, using basic spin quantum states, that P(u,v)=-u*v, where the brackets indicate inner product (dot product).

    now, let A(u,x) denote the predicted value of measuring the spin state of the first electron, and B(v,x) the same for the second. Observe two facts about these functions:

    1. they can only yield the values 1 and -1 (due to the quantized nature of spin)

    2. these are not probabilistic. in the hidden variable schema, the particles have definite values, determined by the hidden variable x. i have indicated this by including x as an argument to the functions.

    Edit: html error, and changed a->u and b->v
     
    Last edited: Nov 13, 2002
  9. lethe Registered Senior Member

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    2,009
    Also note that the antialignment requirement shows that A(u,x)=-B(u,x)

    Using these functions, we can easily calculate P(u,v).

    P(u,v)=Int[\rho(x)A(u,x)B(v,x)]

    to explain that step a little: \rho is the probability density in terms of the hidden variable. basically it just tells us the number of times a particular measurement will occur for a given value of x. it will depend on the as-yet-undetermined way that the determinate values of the spins depend on x. then the expectation value of a quantity (in this case spin-1 along axis u times spin-2 along axis v), is just the value of the quantity, integrated over all states, weighted by the relative likelihood of each state. this is basic probability theory.

    Now to do some math:

    P(u,v)=-Int[\rho(x)A(u,x)A(v,x)], by the anti-alignment requirement.

    Let w be any arbitrary third unit vector. then

    P(u,w)=-Int[\rho(x)A(u,x)A(w,x)], the same as the above equation.

    P(u,v)-P(u,w)=-Int[\rho(x)(A(u,x)A(v,x)-A(u,x)A(w,x))]

    since A(v,x)= 1 or -1, A(v,x)^2=1

    so A(u,x)A(w,x)=A(v,x)^2A(u,x)A(w,x)

    and

    A(u,x)A(v,x)-A(u,x)A(w,x)=A(u,x)A(v,x)-A(v,x)^2A(u,x)A(w,x)
    =A(u,x)A(v,x)(1-A(v,x)A(w,x))

    So
    P(u,v)-P(u,w)=-Int[\rho(x)A(u,x)A(v,x)(1-A(v,x)A(w,x))]

    also, since -1<=A(u,x)A(v,x)<=+1, and 1-A(v,x)A(w,x)>=0 (since A(v,x)A(w,x) is between 1 and -1, 1 minus that stuff must be between 2 and 0).

    so we can change our equality to an inequality and take the absolute value:

    |P(u,v)-P(u,w)|<=Int[\rho(x)(1-A(v,x)A(w,x))]

    here, i lose the A(a,x)A(b,x) term, it can only be less than 1, and i will neglect the sign, since i am taking the absolute value. muliplying by a value less than one decreases the value, so i have my inequality.

    Now, \rho is a probability density, so it integrates to 1, and Int[\rho(x)A(v,x)B(w,x)] = -Int[\rho(x)A(v,x)A(w,x)] = P(v,w) by definition, so my inequality becomes:

    |P(u,v)-P(u,w)|<=1+P(v,w).

    This is the famous Bell s inequality.

    Edit: Arithmetic correction. changed a->u, b->v,c->w
     
    Last edited: Nov 13, 2002
  10. lethe Registered Senior Member

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    2,009
    So, choose u and v to be two perpendicular vectors, and w to be a vector in their plane between them at 45 degrees. then the quantum spin mechanics result i gave at the beginning:

    P(u,v)=-u*v gives
    P(u,v)=0, P(u,w)=P(v,w)=-Sqrt(2)

    Bells inequality says:

    |P(u,v)-P(u,w)|<=1+P(v,w)

    |-Sqrt(2)|<=1-Sqrt(2)

    0.707<=0.293

    Oops! that is obviously not true. this is a proof by contradiction, and the contradiction arises from the assumption of the hidden variable.

    QED

    Edit: fixed html error, and a->u,b->v,c->w
     
    Last edited: Nov 13, 2002
  11. lethe Registered Senior Member

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    2,009
    Comments:

    1. the experiment that einstein used to try to disprove the orthodox position using causality arguments, turned out to be the same experiment that drove the nail in the coffin of the realist position.

    2. The EPR paradox can be resolved if you realize that you cannot actually transfer any energy or signal using the collapse of the wavefunction. it is frequently compared to the motion of a shadow. you can make the motion of a shadow across a screen go as fast as you want (even instantaneously), but you can never transfer energy or momentum or information on a shadow, so it does not violate causality.

    3. bell s theorem holds true only for local hidden variables. some people believe that you can construct a hidden variables theory with nonlocal hidden variables. this is known as Bohmian mechanics (bohm was einsteins student). it may be a workable theory, but in the end, you trade quantum "weirdness" for nonlocality. nonlocality is also weirdness. also, the orthodox theory is much more compact and elegant. bohmian mechanics is a bit unweildy. but, prosoothus, if the orthodox position is obviously false to you, feel free to become a bohmian mechanist. just realize that even if there is such a hidden variable, it does not change the fact that our observations of the real world must follow the heisenberg uncertainty principle, and therefore electrons do not have orbits. this part is experimentally verified, and not open to debate.

    4. schrödinger s paradox is still troubling to some philosphical-minded scientists. i am aware of three ways around this problem.

    i) claim that as soon as the quantum system interacts with a nonquantum system, that is comprises a measurement, and the wavefunction collapses. the cat really is only dead or only alive.

    this is the most common answer, for most scientists who even think about this sort of thing. but most scientists don t think about these sorts of things at all.

    ii) claim it is not a problem. admit that the world is strange, and nature doesn t always show you her whole self. the cat really is in a superposition of dead and alive states.

    a bit spooky, but hey! use your imagination!

    iii)claim that the entire universe exists as one huge wavefunction, and all of the parts of it are connected, and it exists it many superpositions, as many as there are particles in the universe, and that all the quantum possibilities of every event that ever happened are all represented in a different world wavefunction. like there are many versions of the universe at once, one for each of the myriad of possible wavefunctions.

    ideas like the last one are kind of romantic, and have been explored in movies such as 'sliding doors'. it is an interesting concept, but mostly regarded as philosphical sophistry.
     
    Last edited: Nov 23, 2002
  12. lethe Registered Senior Member

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    references: i took most of this from griffiths, with occasional references to bohm.
     
  13. lethe Registered Senior Member

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    i think it should be clear at this point, that if there does seem to be a lack of fundamental understanding of why quantum mechanics is the way it is, it is not for lack of trying.

    if you want to talk about alternatives to the orthodox heisenberg interpretation, you may, but first you must understand the orthodox position inside and out. and i m afraid, prosoothus, that you don t quite understand it that well.

    when i first happened on your thread, i thought you were just someone who wanted to understand how quantum theory works. now it seems much more like you just want to be argumentative, claiming that the theory is wrong when you don t even understand the theory. you will need much more physics before you can claim any theories are wrong.
     
  14. Nasor Valued Senior Member

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    That's easier? LOL.

    Thanks, lethe. Very interesting.
     
  15. chroot Crackpot killer Registered Senior Member

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    lethe,

    These are exactly the kind of posts I have resisted making here on sciforums, due simply to the difficulty inherent in posting mathematical forms.

    Please Register or Log in to view the hidden image!

    Perhaps the easiest way is to type the equations up in Microsoft Word, make them into graphics, and then link in the graphics in your post. In any event, I applaud you for the attempt, and am earnestly waiting to see what kind of questions it provokes.

    And don't mind Prosoothus -- he's one of those types that has already made his mind up about his own correctness long before anything has even been shown to him. He's always right, and we're always wrong.

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    - Warren
     
  16. Prosoothus Registered Senior Member

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    chroot,

    That was supposed to be our little secret.

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    Tom
     
  17. lethe Registered Senior Member

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    yeah, posting math on the web is difficult. making equations with word (or some other equivalent, since ms doesn t make word for my computer) might work, but its kind of ugly. of course, posting math in ascii is also ugly, so... there you go.

    anyway, the point is that there is a mathematical proof of the fact that quantum indeterminacy is incompatible with the statement that the states are determinate, but simply can t be measured. one of those two views is plain wrong.

    the mathematical proof is not too enlightening, so ... anyway, it s there for completeness.
     
  18. Prosoothus Registered Senior Member

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    lethe,

    If you read the fourth post on this thread, you will find my explanation for the EPR paradox. This explanation would also allow for nonlocal hidden variables. What do you think about it??

    Tom
     
  19. Prosoothus Registered Senior Member

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    lethe,

    I'm confused about something you stated earlier in this thread. You said that particles can be in multiple states (or positions) until they are measured.

    What do you mean by measured?? If particle A approaches particle B, isn't particle B being "measured" by particle A, and vice versa??? Doesn't that mean that an electron and a proton in a hydrogen atom have definate values because they are "measuring" each other???

    Tom
     
  20. lethe Registered Senior Member

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    i can t even get started on this. it doesn t even begin to make sense.
     
  21. Prosoothus Registered Senior Member

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    lethe,

    It's relatively simple, and I'd explain it to you but I don't want to hijack your thread with my theory since it is off-topic.

    Tom
     
  22. lethe Registered Senior Member

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    well, thanks for the concern. yeah, i think if you want to talk about how a 0-dimensional point particle is also 6-dimensional (wait a second, didn t you already say that a particle has a location? having a location implies it is a point. a six dimensional particle cannot have a single location in space), then you should do that with a new thread.
     
  23. chroot Crackpot killer Registered Senior Member

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