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- Thread starter Nick V
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ShayanJ

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Your question doesn't seem right to me. If we exclude superposition, even MWI doesn't admit that, because in MWI, different branches of wavefunction occur at different worlds and so in no single world you have two branches happening together. So not only Copenhagen interpretation doesn't allow that, but also MWI doesn't allow it too. In fact no theory should allow it because it causes inconsistency and is not physical.

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In this "minimal interpretation" the wave function (which makes sense only in the nonrelativistic limit; so I'll restrict myself to this limit, which however is already sufficient to understand an astoningishly wide range of phenomena in atomic and condensed-matter physics) describes the state of a single particle as a complex valued function ##\psi(t,\vec{x})##. This function must be square integrable, i.e., the integral

$$N=\int_{\mathbb{R}^3} \mathrm{d}^3 \vec{x} |\psi(t,\vec{x})|^2$$

should be finite. Then one can multiply the wave function by a factor (which is determined only up to a phase factor, which is however irrelevant for the physical meaning of the wave function) such that

$$N=1.$$

Then, according to Born (1926) the modulus squared of the wave function is the position probability density,

$$P_{\psi}(t,\vec{x})=|\psi(t,\vec{x})|^2.$$

I.e., the probability to find the particle in a small volume element ##\mathrm{d}^3 \vec{x}## around the location defined by ##\vec{x}## is ##P_{\psi}(t,\vec{x}) \mathrm{d}^3 \vec{x}##.

Now the wave function can be narrowly peaked around one position. One can prove that there exist such functions with a "width" as small as you want, but there is no state where the width vanishes. This would be a Dirac ##\delta## distribution, but that's not a state because it's not a square-integrable function (even the square itself doesn't make mathematical sense!).

This implies that a quantum particle can never have a precisely determined location. You can give the probability to find the particle in a certain region in space, but never a certain position! This doesn't mean that the particle is at many positions at the same time. Strictly speaking it's even weirder! The particle has no clear position at all although its position can be arbitrarily well determined, i.e., the probability to find it can be very large at some volume small on everyday scales and practically 0 everywhere else; then we say the particle is localized within an uncertainty that is small compared to macroscopic scales or the accuracy of a position measurement.

Further, the quantum mechanical formalism teaches us that not all observables can have sharply determined values. The most famous example, which lead to the Copenhagen interpretation and the idea of complementarity by Bohr are position and momentum. According to quantum theory, a quantum particle cannot have both a quite sharply defined value of the the ##x## component of the position vector and the ##x## component of the momentum vector, but the standard deviations of these quantities, defined with the probability distributions given by any "allowed", i.e., square integrable wave function must obey the famous Heisenberg-Robertson uncertainty relation,

$$\Delta x \Delta p_x \geq \hbar/2.$$

That means: If we have a well-localized particle its momentum distribution is pretty broad and vice versa.

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bhobba

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Most definitely not.

A quantum object is only ever observed to be in one place at a time.

What its doing when not observed is anyone's guess because the theory is silent about that.

Thanks

Bill

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Vanadium 50

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And is untestable, so is not scientific.What its doing when not observed is anyone's guess because the theory is silent about that.

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atyy

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This is not agreed on in general. In Newtonian Mechanics, we can say where an object is when it is not observed. Yet Newtonian Mechanics is generally considered a scientific theory.And is untestable, so is not scientific.

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Vanadium 50

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atyy

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I suspect you are right, and the alternative hypothesis is very beautiful. But I don't know that the alternative is without difficulties. Let's consider an EPR experiment in classical relativistic theory so that spacelike separated measurements are perfectly correlated due to classical correlations prepared at the source (no Bell inequality violation). If Bob assumes that Alice went to Oz when Bob was not observing her, then although he receives Alice's report of her measurement at spacelike separation when they meet up, that report is misleading, since she was in Oz until Bob observed her, and not at the distant spacetime location written in the report.

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As the other comments in this thread and V50's somewhat tongue-in-cheek suggestion make clear, the Copenhagen interpretation does not say that the electron is in two places at once. It says that when we observe it we get one position and that's where the electron is. Obviously that does not preclude the possibility that the electron does all sorts of strange and wonderful things (visiting Oz or Middle Earth, being in two or two thousand places at once, growing legs and dancing, ...) when we aren't watching. But it also gives us no reason to think that any of things are in fact happening.

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bhobba

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Thanks

Bill

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