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## Main Question or Discussion Point

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.

> 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.