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The first lines of the Wikipedia entry Quantum logic give an impression. To cut it short, the essence of the example is this: You have a particle smeared out in some box of length $d$. If you split $d$ into two parts, $d_\text{Left}$ and $d_\text{Right}$, then "the particle is in the union of $d_\text{Left}$ and $d_\text{Right}$" is true by definition, while ...

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The most elementary formulation of Quantum Mechanics (the one usually formulated in Hilbert spaces) can be constructed starting form a lattice of all the elementary propositions which can be tested on a given quantum system obtaining, as the outcome, YES or NOT. This can be similarly done for classical mechanics and, in that case the elementary ...

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Consider for example, a plane vector and two orthogonal unit vectors $\hat x$ and $\hat y$. Any vector in the plane can be expressed as $$\vec v = (\vec v \cdot \hat x) \;\hat x + (\vec v \cdot \hat y) \; \hat y = v_x\; \hat x + v_y\; \hat y$$ So, you're correct, $\vec b \cdot \hat a$ is the component of $\vec b$ in the $\hat a$ direction. And further, ...

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All these functions (and more) are documented in the Handbook of mathematical functions by Abramowitz and Stegun (which is a staple of many professor libraries)

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There are no proofs in physics. PERIOD FULL STOP. There are only measurements and models that are fit numerically. If that is not what you are looking for, you will have to stick with mathematics.

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However then I say: Why do you make things so complicated? Suppose you want to calculate $\exp(A)$. Why don't you define $$\exp(A)~:=~1+A+1/2 A^2 + \ldots$$ and require convergence with respect to the operator norm. An example: Consider the vectorspace spanned by the monomials $1,x,x^2,\ldots$ and let $A=d/dx$. Then you can perfectly define ...

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Try Schaum's Outlines: Quantum Mechanics, ISBN 0-07-054018-7. You'll see the math there, but you'll need to do the deep background studies on all the math from Chapter 2.

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