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Schrodingers Cat and time evolution

  1. May 22, 2008 #1
    There have been countless discussions about the thought experiment about Schrodinger's cat in a box, and how to describe the state of the cat before and after the box is opened, leading to all sorts of bizarre notions like a half dead/half alive cat inside the box, and I certainly don't fully comprehend what it means. It seems to me, however, that in many of the (very rough) mathematical models of the experiment, the time evolution of the system is not fully taken into account. Here's an outline of what I mean:

    > A sample experimental setup: A box perfectly sealed from interaction with the external environment contains a living cat, a flask of poison with a hammer poised above it connected to a geiger counter monitoring, say, a radioactive cesium sample. Say
    that there is a 50% chance that the cesium atom will emit a particle and that it will be detected in a 5 minute interval. The box is sealed, we wait 5 minutes and then open the box. The cat will be alive or dead with a probability of roughly 50%.

    > What happens during that 5 minute interval when the box is sealed and then reopened? Since the box is completely sealed off (by definition) from any interaction with the external universe, it can be considered a closed quantum system.

    > Now, any closed quantum system (think particle in a square well), can be described by a wavefunction representing the state and time evolution of that state based on the Hamiltonian. Let's take a single particle in a 3 dimensional infinite square well. The initial state of the particle can be described by a position wavefunction like

    |psi>= c1|x1,y1,z1>+c2|x2,y2,z2>

    where the x1 (and the other position observables) are eigenfunctions and c1 and c2 are complex numbers.

    > If the particle is initially in a mixed state of different energy levels, as described above, then the position wavefunction will vary with time, being a sum of the eigenfunctions with a factor of exp(iEt/hbar). I believe it would be something like this:

    |psi,t>= c1*exp(iE1t/hbar)*|x1,y1,z1>+c2*exp(iE2t/hbar)*|x2,y2,z2>

    > In the equation above, the position wavefunction will vary with time, sort of undulating between the two eigenstates, but after a time T, the period, it will return to it's initial state. This would still be valid for any initial mixed state, although the period would be longer or shorter depending on the number of terms involved and the initial values.

    > Now if additional particles are added to the box, also in mixed states, these will also follow the same pattern of returning to their initial states, and the overall system of particles will return to its initial state after some value that is a common multiple of the periods of the individual particles periods. For example, if the box has two particles, both in mixed states,
    and the wavefunctions of the particles have periods of T1 and T2 respectively, then in the worst case scenario, the system will return to its original state after a period of T1*T2.

    > Extending this further, if there are N particles in the system, with wavefunctions of periods T1,T2,T3,..,TN, then the entire system will cycle with a period of the least common multiple involving T1,T2,T3,...,TN. Note that this is only true, for a completely isolated quantum system.

    > If the propositions above are true, then any closed quantum system of particles in any mixed state will evolve in time and repeat its cycle of evolution with a period T. Obviously, for large numbers of particles and large numbers of mixed states, the value of T grows very rapidly.

    > Returning to the particles in a box, is it possible that a measurement performed on the system at time t and time t+T, where T is the period of the system would return the same result, since the system would be in an identical state?

    > Finally, extending this to a system such as the cat in the box, if this set of assumptions remains valid, and if the cesium atom could only emit a single particle, then eventually, the system would evolve through three main stages (if the box were never opened)
    (1) Alive, before the counter clicked, t>0
    (2) Dead, after the counter clicked, at some time, t>>0 (this would be a very rapid transition)
    (3) Alive again, where t is almost at the value of T, the period of the entire system (another rapid transition) then back again to (1), and over and over, until the box is opened.

    > From this viewpoint, the opening of the box does not cause a collapse, since the cat would spend nearly all of its time in one of the two essentially macroscopic states, dead or alive, and the transitions of alive/dead and dead/alive would be of the typical duration that it would require poison flask to break and the cat to die, but relative to the period T of the entire cycle of evolution from (1) to (2) to (3) and back to (1), this time interval would be incredibly small.

    > The evolution of the system follows the time-symmetry of standard quantum mechanics, where there is no preferred direction of time, as in all cyclical systems.

    I'm sure there are a thousand holes in this chain of logic and assumptions - I'd really like to hear about them.
     
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  3. May 23, 2008 #2

    ZapperZ

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    Er.. but there ARE experiments in which the time evolution (or time varying) cat-states have been looked at. The Delft/Stony Brook experiments are 2 glaring examples. In these experiments, due to the varying applied field, you can in fact think of the analogous example where the cat fluctuates between dead and alive and all the fractions in between over time.

    So yes, there have been consideration of some form of time evolution of the Schrodinger Cat-states.

    Zz.
     
  4. May 23, 2008 #3
    Thanks ZapperZ - that's very helpful! I'll look for those articles (too bad they aren't on the arXiv or preprint).
     
  5. May 27, 2008 #4
    What about correlated Schrodinger's cats?

    What if we take the Schrodinger cat example one step further and instead of using a radioactive source and detector, use two boxes, two cats and a particle source which produces entangled particles? Here's an outline

    box1 >> cat1 >> detector1 >> (particle source) << detector 2 << cat2 << box2

    I'm not sure how this works, but suppose the particle source works like this - it ejects two spin 1/2 particles at the same time - one to the left sand one to the right. The detectors will measure the spin of the particles on the z axis . Suppose that if the spin is measured down, the poison flask doesn't break and the cat live, but that if the spin is measured up, the flask breaks and the cat dies. Obviously after the entangled particles are emitted and detected, one cat will be dead and one will be alive, and there will now be two entangled cats.

    If this entangled system truly remains quantum and the usual time evolution of quantum systems applies, then eventually, the cats will swap states, with the dead cat returning to life and the live cat dying, and back and forth. I picture the time evolution between the two states as a point on a unit circle, rotating through 360 degrees with a period T the period of the system as a whole, with the very small regions around 0 degrees and 180 degrees representing the extremely short period of superpositions of alive/dead states and the rest of the circle representing the vast array of thermodynamic states which macroscopically would appear as "alive" or "dead" in each box respectively. Nevertheless, as I understand it, the state of every atom and photon in each box would still be correlated throughout the entire process.

    Is this a valid way to think about such a system and could it be mathematically modeled to tell us something useful?
     
  6. May 27, 2008 #5

    StatusX

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    This can't be right by dimensional analysis. In fact, the period will only be finite if T1/T2 is a rational number.

    No. The point about QM is that measurements on identical systems (ie, in the same state) can give different outcomes: we can only predict the probabiliity of getting certain outcomes. We might as well assume the cat/atom/box system is in an energy eigenstate for simplicity, as it doesn't affect the main point of the experiment.
     
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