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Basic Understanding

  1. Dec 20, 2004 #1
    I am old, about to retire, and have been reading some metaphysics books.
    I have not studied physics for 35 years (Feynman diagrams were new then!)
    and I realize that I need help in understanding some basic concepts in
    order to judge these things.

    So, my question:

    This famous Schrodinger's cat situation. You open the "door" into the room but decide to NOT look in. What does the wave equations say now ?
    or
    The double slit experiment with a detector at one slit, sending photons/electorns 1 at a time, no interference. However, if you save the
    information (say a photo of bubble chamber) and then decide to destroy the
    photo's without looking at them, will there be an interference pattern ?

    al
     
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  3. Dec 20, 2004 #2

    selfAdjoint

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    In the cat case, if you don't observe (according to Schroedinger), the wave function doesn't collapse. Doing extra things like opening doors while not observing doesn't count. It's an idealization, a gedankenexperiment, so you can't push it too far.

    In the slit case, once you have observed, that wave function has collapsed (in this case to a single slit). It's a one-way deal, you can't ressurect the wave function by doing something after you observe.

    These answers are based on taking naive observation literally; my own view is that decoherence is a much better approach, even though it is not the final answer in dealing with these issues (I don't know of any final answer).
     
  4. Dec 20, 2004 #3
    Thanks

    I am not familiar with the term "decoherence", after my time ?<g>

    double slit:
    Anyhow, it seems to me that anything that interacts with the photon
    must collapse the wave function even if that is never "observed" by
    the experimenter

    this would seem to suggest that the wave function cannot exist unless
    is has not interacted with anything. But of course, this is not the case
    also since it must be interacting with everything always ???

    obviously, I am confused.

    clarification ?

    al
     
  5. Dec 20, 2004 #4
    Indeed, at the 'heart' of the "Schrödinger Cat Paradox" is the question:

    When does the wavefunction 'collapse'?

    Implicit in the formulation of the paradox is:

    Each of the states "alive" and "dead" can be represented by a 'quantum state'.
    ______________

    Note that, in the 'standard formulation' of Quantum Mechanics, there are two distinct types of mathematical operations which need to be considered. One of these operations has to do with the so-called "wave equation", and that operation is called "unitary evolution". The other operation has to do with the so-called "projection postulate", and it is referred to as "state reduction" (... or "collapse of the wavefunction", etc.).

    Unitary evolution gives us the 'quantum state' of the system as a function of time during a process whereby the 'quantum system' is subject to a ... well – for the moment – let's call it "a non-measuring type of interaction".

    State reduction, on the other hand, has to do specifically with "measurement". Given the type of "measurement" performed (along with the 'quantum state' of the system at the time of "measurement"), this operation allows us to determine the spectrum of all possible results along with the associated probabilities for each possible result.

    ... Tell me, Al Marino, do you have a clear sense of what these two operations are about?
     
  6. Dec 21, 2004 #5
    Not a chance ! If I understand what you mean by "unitary evolution", that is
    what we did back in my day "state reduction" means nothing to me
    A small amount of math may be appropriate here

    I really appreciate your taking the time to discuss this with me.
    I would like to stick to a "possible" experiment scenario with the double slit
    with electrons

    Using a bubble chamber at one slit is, essentially, no different than covering
    up that slit

    So, we need a setup that allows us to tell which slit the electron went thru, works for both slits, one electorn at a time, and allows the electrons to go on to the photo plate

    I don't think Einstein's double slit on springs is possible.
    Is there an experiment that does this ?

    How about a magnetic field ? Do you get interference if there is a magmetic field inserted between slits and photo plate ?

    If not, how about reducing the field strength so low that is does not interfere with the electron (value left as an exercise for the student). Does the interference pattern reemerge ?

    al
     
  7. Dec 21, 2004 #6

    selfAdjoint

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    Al, the evolution of the unobserved wave function is unitary, meaning it conserves probability, the total probability of all the superposed states is and remains 1.

    The "collapse" of the wave function is non-unitary; this is the mathematical correlative of the mystery of the cat.

    Decoherence means that by interacting with the rest of the universe the wave function evolves toward classical states; in the math that means the matrix of amplitudes (complex precursors of probability) evolves toward diagonal.
     
  8. Dec 21, 2004 #7
    I have no experience calculating wave function/prob. for a wave function that does not interact with anything. When it interacts or decays, you can calc prob of what happens and when (if I remember correctly)

    What causes the "collapse" of wave function, and why is it non-unitary ?

    What does "interaction with the rest of the universe" mean and how does it lead to diagonal values (does this refer to Heisenberg's matrix formulations ?)

    I think it best to point me to a good book for this (does Hallday & Resnick have something on this ?)

    What I thought I could get answered here was the question of interference in an actual experiment (don't tell me, you have been trying to tell me <g>)

    Is it possible to set up an experiment as I suggested ?
    What are the results ?
    Surely, in 80 years, all possible experiments have been done

    al
     
  9. Dec 21, 2004 #8

    reilly

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    Fundamenally, the Schrodinger cat paradox has nothing to do with QM, and everything to do with probability. To wit; the immortals, Tom and Bob (first met in 44th grade arithmetic class) agree initially walk in opposite directions, say along Madison Ave. in New York. They start at 52nd St. When Bob gets to 54th, he goes straight on Madison if he gets a green light, and goes right toward 3rd Ave. He keeps going straight unless forced by red lights or barriers to turn right.

    Tom does much the same thing. When he gets to 50th, he goes straight or right, depending...

    After 1/2 hour or so they talk, via cell phone, and explain where they are and how they got here. Suppose they each made 10 binary decisions, which gives 2**10 paths. If you insist upon the 1/2 alive, 1/2 dead approach for the cat. So, do you not have to make a convoluted set of allocations for Tom and Bob -- they are simultaneously in all of the 2**20 the paths. Further, the reduction process, via the cell phone, has its own probability structure -- what will the sequence of discussion and discovery be? So we'll end up with more states than imaginable, that supposedly all have reality(?)

    A classical probability perspective says: the Tom-Bob system can go through many possible states during 0 - t, say. There is no reason to ascribe some form of reality to each of these states during the experiment -- no need for 1/2 dead, 1/2 alive. if you do, for what possible purpose? Why go to such lengths to distort what we do know about reality?. Ogden Nash noted that purple cows do not exist. What in the world is a 1/2 dead. 1/2 alive cat, relative to say a 100% alive one? like the Chesire cat, the 1/2-1/2 cat somehow vanishes when seen, only to be replaced by a 100% dead or 100% alive at.

    What's wrong with "We don't know till we look? Why do we need this odd creature, a 1/2 dead, 1/2 alive cat? Are we talking reality or metaphor?

    And, for another time, there's the approach that says. QM allows us to compute probabilities, state descriptions and the like. In short, the wave function collapse is simply the collapse from uncertainly to certainty in our brain. (Sir Rudolph Peierls favors this approach.)

    Regards,
    Reilly Atkinson
     
  10. Dec 21, 2004 #9
    I asked Al, regarding the operations of "unitary evolution" and "state reduction":
    ... and the response was:
    I hesitate to begin with a double-slit scenario, because it will involve extraneous complexity which may obscure the basic features of "unitary evolution" and "state reduction".

    So, let's begin with something simple.

    To highlight the distinction between the two operations of "unitary evolution" and "state reduction", consider a quantum system whose 'state space' is 2-dimensional. Suppose we will eventually perform a "measurement" on the system that determines which of the following two mutually exclusive alternatives has prevailed:

    (i) the system is found to be in the state |1> ;

    (ii) the system is found to be in the state |2> .

    Items (i) and (ii) tell us the type of "measurement" which will be performed, and the vectors |1> and |2> are necessarily orthogonal.

    So, how does the analysis go?

    Well ... first, start off with an initial 'quantum state', say |ψo>. Then, invoke "unitary evolution". This means we find the solution |ψ(t)> to the Schrödinger equation subject to the initial condition |ψ(0)> = |ψo>. This time evolution is called "unitary" because the operator U(t',t) which takes |ψ(t)> to |ψ(t')> (i.e. |ψ(t')> = U(t',t)|ψ(t)>) is itself a unitary operator (... and this can be proven from the form of the Schrödinger equation).

    Now great! We know the 'quantum state' of the system at any time t. Let us write this state in terms of the basis {|1>,|2>}; i.e.

    |ψ(t)> = a(t)|1> + b(t)|2> .

    And now, at time t, let us perform a "measurement" of the type described by mutually exclusive alternatives (i) and (ii) above. Then, "state reduction" tells us following:

    Prob(t)[system is found to be in state |1>] = |a(t)|2 ,

    and

    Prob(t)[system is found to be in state |2>] = |b(t)|2 .

    In the case of a "filtering-type" of measurement (also called a "nondemolition" measurement) in which the 'quantum system' is preserved in its measured state, we then have a transition at time t whereby

    |ψ(t)> → |1> , with probability |a(t)|2 ,

    or

    |ψ(t)> → |2> , with probability |b(t)|2 .


    ... I think I will stop here, and let you ponder this for a while. I too will have to think about it, because (it seems that) all of the questions you are asking can be formulated relative to the above framework. On the other hand, it appears that you would like to see your questions answered relative to a specific and clear-cut experimental setting like the double-slit.

    In any case, at least some of your questions ought to be brought to the fore in light of the above. And that is one of the things I would like to think about.
     
  11. Dec 22, 2004 #10

    DaveC426913

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    "What's wrong with "We don't know till we look? Why do we need this odd creature, a 1/2 dead, 1/2 alive cat? Are we talking reality or metaphor?"

    The cat really is in both states. It's not just probability about which.

    (Note that it's not the cat that causes the 'two-state'ness, it's the subatomic particle that both did and did not get emitted from the radioactive source. The cat is merely a macroscopic consequence of a subatomic event.)

    This is the fundamental paradox of quantum mechanics. It shows that the state of the particle has not been defined until it is observed.
     
  12. Dec 22, 2004 #11
    Dave,
    The question is really "what the hell does observed mean"

    Eye in the Sky
    (by the way, why that moniker ? are you currently on the space shuttle)

    Your fomalism appears (to me anyway) to be adequate
    I now have some understanding of the terms "unitary evolution" and "state reduction"

    However, your |ψ(t)> → |1> , with probability |a(t)|2
    does not answer what interaction would cause the reduction and what would NOT
    (the interaction itself may have a function that includes the Dirac delta function, but unless we know what it is, how does that help?)

    lets look at an experiment !

    al
     
  13. Dec 22, 2004 #12

    Doc Al

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    I don't believe there is any compelling reason to treat the cats as ever being in a superposition of states (half dead, half alive). As soon as the Geiger counter clicks (or doesn't) one can assume that sufficient decoherence has taken place to change the initial superposition into a classical mixed state. There is no reason whatsoever to think that the act of a person looking into the box has any effect. (Yes, some people do think that--but I don't think such a view is justified.)
    Same basic point. As soon as an irreversible macroscopic change has occurred, the initial state can be considered to have decohered into a mixture thereby destroying any interference pattern. It doesn't matter whether a person looks at the data or not.

    Now as to exactly how decoherence works, don't ask me. :smile: That's an active area of research.
     
  14. Dec 22, 2004 #13
    Doc,
    Thanks for reply.

    <As soon as an irreversible macroscopic change has occurred, the initial state can be considered to have decohered into a mixture thereby destroying any interference pattern. It doesn't matter whether a person looks at the data or not.>

    This seems logical to me also. The rub is exactly what constitutes an "irreversible change"

    To repeat some of my earlied post

    Using a bubble chamber at one slit is, essentially, no different than covering
    up that slit

    So, we need a setup that allows us to tell which slit the electron went thru, works for both slits, one electorn at a time, and allows the electrons to go on to the photo plate

    I don't think Einstein's double slit on springs is possible.
    Is there an experiment that does this ?

    How about a magnetic field ? Do you get interference if there is a magmetic field inserted between slits and photo plate ?

    If not, how about reducing the field strength so low that is does not interfere with the electron (after all, this must be quantized also). Does the interference pattern reemerge ?

    Do you know of any experiments that do this ?

    al
     
  15. Dec 23, 2004 #14

    reilly

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    Dead/alive cats

    Before getting to QM, I'll note that, in practice, your fundamental paradox applies equally well to classical stochastic processes, c.f. Kalman Filters. and all that stuff.



    The cat most certainly is not in both states, and, if it were, what do you gain by such a situation? I'll cite a couple of books you should read if you truly wish to understand the complexities of QM -- Kemble's Quantum Mechanics, available from Dover, which discusses QM measurements in great detail, and demonstrates the reasonableness of the knowledge interp. of QM(Again, the nobelist Sir Rudolph Peierls has written about this take on QM) See also Einstein, Bohr, and the Quantum Dilemma, A. Whittaker , a thorough history of the interp of QM.

    If you do a Hume on this reality ascription, you'll drive yourself crazy. Take your automobile, or cell phone or whatever. if you probe a bit, some of your friends will have different notions about the future of thos items. John says the car will die ned week, Fred says it will go until your toaster blows up. Roscoe, that devilish guy, thinks a lot about getting some radioactive stuff, and, maybe, if he feels up to it. sorta' doing a Schrodinger cat thing, if he can find a cat, and a suitable room. With a serious belief in your 1/2 -1/2 alive -dead, how in the world do you construct the reality of a chain of conditionals, that can be shifting, odds change. What's the point? (I'd be very interested to see the basic justification for your position.)
    Regards,
    Reilly Atkinson
     
  16. Dec 23, 2004 #15
    ... JACKPOT!!!

    In the 'standard formulation' of Quantum Mechanics, there is NO representation of "measurement" as a dynamical process. In that formulation, any dynamical process has an associated Hamiltonian and is therefore characterized by "unitary evolution". But a "measurement process" such as that above in which the quantum state

    |ψ> ≡ a|1> + b|2>

    evolves according to

    |ψ> → |1> , with probability |a|2 ,

    |ψ> → |2> , with probability |b|2 ,

    cannot be achieved by a "unitary" transformation – or 'RESTRICTION' thereof – of any kind ... even when such a prospective unitary transformation is permitted (as one suspects it must) to involve Hilbert spaces corresponding to "measuring apparatus" and "environment", followed by subsequent 'RESTRICTION' ("partial tracing") to the Hilbert space of the said 'quantum system' alone.

    This, of course, is an essential aspect of the notorious "MEASUREMENT PROBLEM" of Quantum Mechanics. (It should be noted that not all 'formulations' of QM are plagued by such a problem (e.g. Bohm's interpretation)).

    To put it much more simply ... in the 'standard formulation' of Quantum Mechanics, the Dynamical Postulate (involving the Schrödinger equation) and the Projection Postulate (giving the rule for "state reduction") appear as two equally necessary, yet formally independent, axioms.

    ... Again, I will stop here and let you ponder the above, as I too think about it more myself.


    In the mean time, I will leave you with the words of Bernard d'Espagnat on the above ideas:

     
  17. Dec 23, 2004 #16
    Eye in the Sky
    I am beginning to get an inkling here; thanks (now I remember why I liked physics so much a long time ago)

    It seems reasonable that any formulation must include the measuring apparatus
    However, I'm not clear about your reference to Hilbert spaces other than they can include discontinuous functions (besides, it hurts too much to think of them)

    "Standard formulation" implies something other formulation; what would that (they) be?
    (I will look up Bohm, who I have never heard of)

    <To put it much more simply ... in the 'standard formulation' of Quantum Mechanics, the Dynamical Postulate (involving the Schrödinger equation) and the Projection Postulate (giving the rule for "state reduction") appear as two equally necessary, yet formally independent, axioms.>

    Does this imply I can say (to paraphrase) "LET THERE BE RESOLUTION !" <g>

    This guy Bernard d'Espagnat seems to put his finger on the problem. I will try to look him up too.

    Perhaps you can point me in the right direction to learn more about the "measurement problem" and the experimental attempts to resolve or get around it.

    I thank you again. I hope you have a very Merry Christmas and New Year

    al
     
  18. Dec 24, 2004 #17
    A [pure] 'quantum state' (of a 'quantum system') is a vector in a Hilbert space.

    A "Hilbert space" is essentially a vector space with an inner product ... and for the finite-dimensional case this description is complete – there is no more to be said. For the infinite-dimensional case we need to add two more properties to the list, namely, "completeness" (as part of the formal definition) and "separability" (as a further condition) [note: for the finite-dimensional case both of these conditions hold by 'default']. In practice, however, there is no need at all to delve into the nuances of these two mathematical properties, because we can simply define an infinite-dimensional separable Hilbert space as follows:

    It has a countably-infinite orthonormal basis {e1,e2,...} and it consists of all of the "objects" (i.e. "vectors")

    nαnen (where each αnЄ C)

    such that

    nn|2 < ∞ .

    Formally, this is quite simple ... is it not?
    -------
    ... And here is a concrete example:

    Consider the vector space of square-integrable functions on the real interval (0,L). Then, the functions

    φn(x) = √(2/L) sin knx , n = 1,2,3,... ,

    where kn = nπ/L ,

    form a countably-infinite orthonormal basis with respect to the inner product defined by

    (f,g) ≡ ∫0L f*(x)g(x) dx .

    Any square-integrable function f on (0,L) can be written as

    f = ∑nanφn ,

    where anЄ C, and

    n|an|2 < ∞ .

    [note: in the above, any pair of functions f and g such that

    0L |f(x) - g(x)|2 dx = 0

    are considered to be one and the same 'function']
    -------
    ... I hope that the above remarks have not been cause for aggravation.
    ___________
    ... include the measuring apparatus HOW?

    In (what I am calling) the 'standard formulation' of QM, the measuring apparatus does not appear in the mathematical formalism on equal "footing" with the quantum system. As mentioned above, a [pure] 'state' of the latter is a vector in a Hilbert space ... whereas, a 'state' for the measuring apparatus has NO representation whatsoever in the mathematical formalism! The apparatus enters into the formalism as something which, so to speak, "partitions" the Hilbert space (of the 'quantum system') into mutually orthogonal subspaces (... and this is a fixed "partition", according to the type of measurement which the apparatus performs), and it is only (any) one of these subspaces into which the "state reduction" can occur.
    ___________
    When I use the term 'standard formulation', I mean the usual undergraduate-textbook type of presentation along with the associated elements of interpretation. These are essentially of the 'von Neumann' genre. Thus, the (implied) "something other" 'formulations' would then include all manners of approach which deviate from the said 'standard'. Most of these "other" 'formulations' do not modify the Schrödinger equation, and they are more commonly referred to as "interpretations". Regarding those approaches which do modify the Schrödinger equation, I am largely unfamiliar with them, but I have seen such descriptive terms as "stochastic extension of the Schrödinger equation" and "nonlinear state space".
    ___________
    In my view, relative to the 'standard formulation', the answer is a definitive "YES!".

    However, it must be emphasized that, to many practicing physicists the "RESOLUTION" is thought to be quite a modest one. For example:
    And from another thread in this forum ("Schrödinger's Cat", post #9):
    You may wish to note the rebuttals there in posts #10, #11, and #13.

    In my view, however, none of those rebuttals are successful (... and indeed, post #13, in particular, is not even directed) at refuting the contention that

    a "knowledge-based approach" to the 'collapse' phenomenon can be consistent with the QM formalism .

    Indeed, the above contention must be true since "Bohm's interpretation" is an instance of it.

    On the other hand, the validity of this contention does not negate that there must be direct "physical reality" of some kind ascribable to the wavefunction. In Bohm's interpretation this can be seen most explicitly, for, there, the wavefunction itself is what gives rise to the so-called "quantum potential".
    ___________
    As possibilities come to mind, I will let you know. It would, however, be helpful if you could indicate the level of mathematics which you are comfortable with. ... For example, how much of the above with regard to the "Hilbert space" were you able to follow?

    By the way, why do you say "experimental attempts to resolve or get around it" ? – it is a problem which concerns the theory.
     
    Last edited: Dec 24, 2004
  19. Jan 1, 2005 #18
    Eye_in_the_Sky,
    I've been able to follow your brief presentation of Hilbert Spaces, although I'm only a poor engineer. I'm very curious to read more on the topics of QM interpretation (not necessarily HS). BTW, your explanation of the Dynamical Postulate as application of a unary operator and the Reduction Postulate as application of a projection operator is the clearest I have ever read. I guess the unary operator you are talking about is simply the time evolution operator associated to the Schrodinger equation - please correct me if I'm wrong. Anyway, thanks a lot for sharing!
    Back to my question: what book or articles could I read to understand more?

    Francesco
     
  20. Jan 10, 2005 #19
    To Al and Francesco

    With regard to "quantum measurement", I think a good place to start is

    http://en.wikipedia.org/wiki/Quantum_measurement .

    Towards the bottom of that page, there is a list of additional topics (under the "See also" title).


    Another good resource is

    http://plato.stanford.edu/contents.html .

    But the articles there can be quite difficult.

    Here are search results for the query "quantum measurement":

    http://plato.stanford.edu/cgi-bin/w...s=30&maxchars=10000&query=quantum+measurement

    Here are search results for the query "interpretation of quantum mechanics":

    http://plato.stanford.edu/cgi-bin/w...000&query=interpretation+of+quantum+mechanics


    I have only read a few layman oriented books on Quantum Mechanics and its interpretations, the best of which I found to be "Quantum Reality, Beyond the New Physics" by Nick Herbert.
     
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