Dear Folks,

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


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 
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 | 
Note that the Sum_{n} |<e_n | psi>|^2 = 1 since |psi> is a vector of unit 

This description shows that quantum theory is a dynamic sort of probability 
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 
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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