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Math Help - Understanding Schrodinger's Cat

  1. #1
    Senior Member Paze's Avatar
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    Understanding Schrodinger's Cat

    So I want to understand Schrodinger's Cat but obviously it's very vague to just watch a video or read an excerpt on the thought experiment itself. I want to actually understand why the mechanics behave as they do.

    I've watched the Slit experiment and I find it interesting but what other theories/experiments is/are a good supplement to understand Schrodinger's Cat? Thanks.
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    Member Ruun's Avatar
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    Re: Understanding Schrodinger's Cat

    Schrödinger equation can be understood as a linear, second order, partial differential equation

    i\hbar \frac{\partial \Psi(\vec{x},t)}{\partial t}=-\frac{\hbar^2}{2m}\nabla^2\Psi(\vec{r},t)+V(\vec{r  },t)\Psi(\vec{r},t)

    where \nabla^2\Psi(\vec{r},t)=\frac{\partial ^2 \Psi(\vec{r},t)}{\partial x^2}+\frac{\partial ^2 \Psi(\vec{r},t)}{\partial y^2}+\frac{\partial ^2 \Psi(\vec{r},t)}{\partial z^2}

    There exists other interpretations of this equation as a first order linear equation in time for an element \Psi(\vec{r},t) on a Hilbert Space \mathcal{H}. In this context, you can define a linear operator called Hamiltonian

    \hat{H}=-\frac{\hbar^2}{2m}\nabla^2+V(\vec{r},t)

    And re interpret the Schrödinger equation as

    i\hbar \frac{\partial \Psi(\vec{r},t)}{\partial t}=\hat{H}\Psi(\vec{r},t)
    .

    It has great interest to discuss the case V(\vec{r},t)=V(\vec{r}), this is, time independent potentials. Then a particular family of solutions can be "easily" found by imposing that

    \Psi(\vec{r},t)=\psi(\vec{r})T(t)

    As in the usual examples of separation of variables, we will found that a separation constant is needed, call it
    E
    .. that can be interpreted as the energy of the system. Among other things it will happen that

    \Psi(\vec{r},t)=\psi(\vec{e})e^{-iEt/\hbar}

    And a very interesting feature (depending on the potential V(\vec{r})) is that there may be more than one separation constant, it may be happen that there exists even an infinite denumerable set \{E_{n}\} that solves the equation, so you just can't drop some of them, and you will have to take the sum or even series.

    Now, to the cat. Suppose that you're given the Schrödinger cat potetial V_{cat} and that the most general solution is

    \Psi=\psi_{alive}+\psi_{dead}

    This is the solution. The system is neither alive nor dead, it is in both systems at the same time. Now, another axiom of Quantum Mechanics is that when you measure the quantity deadness or liveness of the cat it will be either alive or death, not both of course. But after measuring the state is a linear superposition of both states. So the Schrödinger cat is a very poor choice of words to replace linear superposition. Which of course doesn't happen in classical mechanics, and this is why it is wierd. But I'm afraid that Nature is quantum-mechanical, strange for us or not.
    Thanks from Paze
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    Senior Member Paze's Avatar
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    Re: Understanding Schrodinger's Cat

    That is a little out of my league, but thank you
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    Re: Understanding Schrodinger's Cat

    You want to understand Schrodinger's Cat? Place a cat in a box of poison and close the lid. Until you let the cat out, you don't know if the cat is alive or dead, and so can be considered to be in both states at once.

    The moral of the story - don't live in doubt about anything, it's better to know for sure
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    Senior Member Paze's Avatar
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    Re: Understanding Schrodinger's Cat

    What if you have a video camera inside the box and YOU know whether the cat is dead or not. Is the cat still in a superposition relative to me? :/
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    Re: Understanding Schrodinger's Cat

    If the cat is unconscious, you would not be able to tell if it was dead or alive, even with a video camera...
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    Senior Member Paze's Avatar
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    Re: Understanding Schrodinger's Cat

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    Member Ruun's Avatar
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    Re: Understanding Schrodinger's Cat

    There are many problems with quantum mechanics. It is a very beatiful framework and extremely successful in order to describe systems in a scale comparable to their de-Broglie wavelength (which is the natural scale for non-relativistic quantum mechanics, being the Compton wavelength the natural scale for relativistic quantum mechanics, so to speak) but as far as that realm is not part of our everyday experience, we fail in order to interpret that framework.

    Of course you can do quantum mechanics without interpreting the theory, or, as many physicists do (myself included) subscribe to some short of Copenhagen-like interpretation and not worry too much about it.
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