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 Wave function reduction and evolution 
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Joined: 06 Jan 2014, 11:17
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Post Wave function reduction and evolution
In the book was expressed a hope for a consistent description of wave function evolution and reduction. When trying to understand EPR paradox, it occurred to me, that reduction is a consequence of entanglement of the observer with the observed object.

Let's consider two photons of the EPR paradox: \frac{1}{\sqrt{2}} \left | \varphi_{12} \right \rangle + \frac{1}{\sqrt{2}} \left | \varphi_{21} \right \rangle - polarized in opposite directions with equal probability. And the observers \psi_{1} and \psi_{2}, observing the first and the second photons respectively, their evolutions:
\textbf{U} \left | \psi_{1} \right \rangle \left | \varphi_{12} \right \rangle = \left | \psi_{11} \right \rangle \left | \varphi_{12} \right \rangle
\textbf{U} \left | \psi_{1} \right \rangle \left | \varphi_{21} \right \rangle = \left | \psi_{12} \right \rangle \left | \varphi_{21} \right \rangle
\textbf{U} \left | \psi_{2} \right \rangle \left | \varphi_{12} \right \rangle = \left | \psi_{22} \right \rangle \left | \varphi_{12} \right \rangle
\textbf{U} \left | \psi_{2} \right \rangle \left | \varphi_{21} \right \rangle = \left | \psi_{21} \right \rangle \left | \varphi_{21} \right \rangle
\psi_{11} - the eventual state of the first observer after he measured photon 1 with polarisation 1, \psi_{12} - for polarisation 2. So for both observers to measure polarization 1 would be for photons state \varphi_{11}, which doesn't exist.

This experiment also lets us see, how the observers wave functions evolve, and we can see, how they see wave function reduction.
When the observer begins the measurement, we have the total wave function:
\left | \psi_{1} \right \rangle \left ( \frac{1}{\sqrt{2}} \left | \varphi_{12} \right \rangle + \frac{1}{\sqrt{2}} \left | \varphi_{21} \right \rangle \right ) = \frac{1}{\sqrt{2}} \left | \psi_{1} \right \rangle \left | \varphi_{12} \right \rangle + \frac{1}{\sqrt{2}} \left | \psi_{1} \right \rangle \left | \varphi_{21} \right \rangle
Because of the linear nature of the Schrödinger equation and independence of components \varphi_{12} and \varphi_{21}, components of this linear combination of wave functions evolve independently, but notice that when \left | \psi_{1} \right \rangle \left | \varphi_{12} \right \rangle evolves independently, we know the result: it's just an observer measuring state \varphi_{12}, the result of evolution is \psi_{11} - he will measure photon 1 in state 1 as if there is no other components of the state, theoretically the photons state is \frac{1}{\sqrt{2}} \left | \varphi_{12} \right \rangle + \frac{1}{\sqrt{2}} \left | \varphi_{21} \right \rangle, but he sees only \varphi_{12} and no trace of \varphi_{21} state - the wave function collapsed.

So the evolution is as follows:
\textbf{U} \left | \psi_{1} \right \rangle \left ( \frac{1}{\sqrt{2}} \left | \varphi_{12} \right \rangle + \frac{1}{\sqrt{2}} \left | \varphi_{21} \right \rangle \right ) = \frac{1}{\sqrt{2}} \textbf{U} \left | \psi_{1} \right \rangle \left | \varphi_{12} \right \rangle + \frac{1}{\sqrt{2}} \textbf{U} \left | \psi_{1} \right \rangle \left | \varphi_{21} \right \rangle
= \frac{1}{\sqrt{2}} \left | \psi_{11} \right \rangle \left | \varphi_{12} \right \rangle + \frac{1}{\sqrt{2}} \left | \psi_{12} \right \rangle \left | \varphi_{21} \right \rangle

Whatever result he gets, he sees only one result and no trace of the other, so in both states he suggests that wave function reduction happened.

This also suggests that a non-destructive measurement is impossible, because when the observer receives information about the object state, he has no way but to become entangled with the object state component, which supplies the result of measurement.

Well, the multiverse interpretation takes it a little too far, this seems like just a superposition of quantum states, but it's probably still a good intuitive way of reasoning about quantum mechanics.

31 May 2014, 14:01
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