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Quantum Mechanics: Just in case you wondered–

August 18, 2013

1.2 Basic axioms of quantum mechanics…

Axiom (States). The state of a quantum mechanical system is given by a vector in a complex vector space H with Hermitian inner product …. *

(* Is this the quantum version of a happy face? Can a Hermitian inner product ever be happy?)

Note two very important differences with classical mechanical states:

The state space is always linear: a linear combination of states is also a state.

The state space is a complex vector space: these linear combinations can and do crucially involve complex numbers, in an inescapable way. In the classical case only real numbers appear, with complex numbers used only as an inessential calculational tool….
[T]he notation introduced by Dirac for vectors in the state space H: such a vector with a label ψ is denoted:|ψ⟩

Axiom (Observables). The observables of a quantum mechanical system are given by self-adjoint linear operators on H….

Axiom (Dynamics). There is a distinguished observable, the Hamiltonian ℋ. Time evolution of states |ψ(t)> ∈ H is given by the Schrödinger equation: d/dt(|ψ(t)⟩) = − (i/ħ)(ℋ|ψ(t)⟩)

The Hamiltonian observable ℋ will have a physical interpretation in terms of energy, and one may also want to specify some sort of positivity property on ℋ in order to assure the existence of a stable lowest energy state. ħ is a dimensional constant, the value of which depends on what units you use…. We will see that typically classical physics comes about in the limit where(energy scale)(time scale)/ħis large….

Principle (Measurements). (1) States where the value of an observable can be characterized by a well-defined number are the states that are eigenvectors for the corresponding self-adjoint operator. The value of the observable in such a state will be a real number, the eigenvalue of the operator. (2) Given an observable O and states |ψ1⟩ and |ψ2⟩ that are eigenvectors of O with eigenvalues λ1 and λ2 (i.e. O|ψ1⟩=λ1|ψ1⟩ and O|ψ2⟩=λ2|ψ2⟩), the complex linear combination state c1|ψ1⟩ + c2|ψ2⟩ may not have a well-defined value for the observable O. If one attempts to measure this observable, one will get either λ1 or λ2, with probabilities c1^2/(c1^2 + c2^2) and c2^2/(c1^2 + c2^2), respectively.

This principle is sometimes raised to the level of an axiom of the theory, but it is better to consider it as a phenomenological over-simplified description of what happens in typical experimental set-ups…
Peter Woit: Quantum Mechanics for Mathematicians http://www.math.columbia.edu/~woit/QM/fall-course.pdf

Now you know — or you don’t know. It is quantum mechanics after all. On the other hand, perhaps it is all a phenomenological over-simplified description of what happens. Wouldn’t that be a shame?

You know now that I think about it, God seems to be a simpler source for it all than quantum mechanics. Perhaps it is as Kurt Goedel implies, no matter how far you go or how deeply you think about it, sooner or later it all comes down to six of one or a half-dozen of the other.

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