This interpretation of quantum mechanics is named after the city where its creator Niels Bohr worked and lived. It is the principle of complementarity taken a step further. 

Trying to explain the results from the double-slit experiment/wave/particle duality are hard in our thought world. The view of Bohr was that theories should be strictly based on experimental observations, and not inhibited by limitations in our philosophical system and imagination. The indeterminacy of said experiments and Heisenberg's uncertainty principle for instance should be seen as real and physical, and not the effect of imperfections of the theories.

This interpretation is what had Albert Einstein say

God doesn't play dice

but dear old Einsten was in fact proven wrong. God does indeed play dice. See Einstein-Podolsky-Rosen paradox for more on that.

One famous theory that follows the Copenhagen interepretation is the one that there are many universes. It was introduced by John Archibald Wheeler in 1957. His idea was that everytime a choice is made on the microscopical level, the universe is split up so that every possible occurence actually happen in one universe. Or to put it in the Schrödinger's Cat scenario: In one universe that cat is alive, in one it is dead.

There are three views (this was the way it was presented to me, anyway) regarding the interpretation of quantum physics, and what the wavefunction really means.

In quantum mechanics, the wavefunction is all that describes a particle. Simply put, it allows one to determine the probability of a particle being at a given point. Only a probability, mind you, not anything certain.

So, you measure the location of the particle. You get a measurement. (This is collapsing the wavefunction). Now, you know at the time of your measurement, your particle was at location x. But where was the particle right before you measured it? The answer to this question split theorists into three groups, each with their own view:

The realist view: The particle was at point x right before you measured it as well. This would make sense, from a normal-physics point of view. But if this was true, quantum mechanics wouldn't be a complete theory, since it couldn't predict for sure that the particle was at x when it really was. Therefore, some other information in addition to the wavefunction would be needed for the complete picture. This is what Einstein believed. It would mean, in essence, that there really isn't any indeterminacy, we just haven't figured it all out yet.

The Copenhagen interpretation(the orthodox view): The particle really wasn't anywhere. Only by measuring have we forced it to assume a location. As stated above, this view is associated with Niels Bohr, and has been the most widely accepted position. This means the indeterminacy is real, and not just something we can't get rid of yet

The agnostic view: i.e. Don't answer. It makes no sense to talk about where the particle was before the measurement, because the only way you'll know where it is, is by doing the measurement. You can't physically determine anything about 'before you measure', so the question isn't physically meaningful

That's how things stood until 1964. Each view had their supporters, and there was no way to tell which view was the correct one. But then, in 1964 John Bell discovered that it did in fact make an observable difference whether the particle had a precise (even if unmeasurable) position before the measurement. This removed agnosticism as a choice, since the answer did in fact influence the results of experiments.

Since then, experiments have confirmed that the realist interpretation cannot be true, at least in its original form. This leaves the Copenhagen interpretation, though unproven, as the majority opinion. Some alternate hypothesis (like the many worlds interpretation mentioned above) are still possible, and a few other loopholes also remain.

But all in all, it currently seems that Einstein was wrong, and indeterminacy is a real, physical fact of nature.


Addenum: It's been noted elsewhere that while Bell's experiment does rule out local deterministic theories, it does not rule out nonlocal deterministic theories. The differences in those two I don't feel qualified to node.

This information gleaned from and partially paraphrased from Introduction to Quantum Mechanics by David J. Griffiths

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