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The Uncertainty Principle

Here is a brief explanation of why the "Heisenberg uncertainty principle" is implied by quantum theory. It's not nearly as mysterious as people think. Yes, if you don't have any matrix math background you won't be able to follow this. But you still might see it's just a little bit of math, there isn't a lot of stuff to it. The Uncertainty Principle is not a Principle, it's not a law of physics, it's just one of many results you can work out about quantum theory with a small amount of math:

-- In quantum theory observables can be represented by Hermitian matrices.

-- If an observable of a system can be represented by a particular matrix at a particular instant, then all matrices of the same dimension represent observables of that system.

-- In a state specified by the vector |psi>, an observable X is sharp if and only if X|psi> = x|psi> for some real number x. In which case x is an eigenvalue of X and |psi> is an eigenvector of X.

Now let Y be any matrix that does not have |psi> among its eigenvectors. (For any vector, there exists an infinity of such matrices.)

If the actual state is |psi>, the observable Y cannot be sharp. (Because of the 'if and only if' above.)

Elliot Temple on March 5, 2015

Comments (1)

In other words, not all observables of a system are sharp at any given point in time. (Sharp means the same in all universes.)

curi at 5:58 PM on September 4, 2017 | #8982

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