Dear Folks, I am sending this again, just the quantum part, with typos removed.

Best, Lou Quantum Theory in a Nutshell 1. A state of a quantum system is a vector |psi> of unit length in a complex vector space H. H is a Hilbert space, but it can be finite dimensional. Dual vectors are denoted by <phi | so that <phi |psi> is a complex number and <psi |psi> is a positive real number. 2. A quantum process is a unitary transformation U: H ——> H. Unitary means that the U* = U^{-1} where U* denotes the conjugate transpose of U. Unitarity preserves the length of vectors. 3. An observation projects the state to a subspace. The simplest and most useful form of this is to assume that H has an orthonormal basis { |e_1> ,|e_2>,…} that consists in all possible results of observations. Then observing |psi> results in |e_n> for some n with probability |<e_n | psi>|^2. Note that the Sum_{n} |<e_n | psi>|^2 = 1 since |psi> is a vector of unit length. This description shows that quantum theory is a dynamic sort of probability theory. The state vector |psi> is a superposition of all the possibilities for observation, with complex number coefficients. Via the absolute squares of these coefficients, |psi> can be regarded as a probability distribution for the outcomes that correspond to each basis element. Since the coefficients are complex numbers and the quantum processes preserve the total probability, one has room for complexity of interaction, phase, superposition, cancellation and so on.

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