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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