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Sketch:
Energy Spectrum of the Quantum Theoretical Oscillator

Symbols
like "*a*" or "*b*" can be concatenated
or juxtaposed, giving the prototype of a multiplication. Disrespecting parenthization
*(ab)a=a(ba)=aba* is called
associativity of the multiplication. Respecting the order of the symbols
*ab* and *ba* makes the
multiplication noncommutative. (In mathematics the emerging structure is
called a free monoid generated by the elements "*a*" and "*b*". )

The
monoid elements can be written as being added by grouping them around a "plus"-sign.
And a monoid element can be written as being multiplied by a number (complex
or real) by prefixing it with the given number. This is the basis for
realizing the Heisenberg commutation-relation with monoids (The one represents
the neutral element of the monoid-multiplication.):

Iterations
over the natural numbers give:

And
the energy eigenvalue problem of the one-dimensional harmonic oscillator
then can be formulated as follows:

That
problem is solved by using extensions of linear combinations to infinitely
many summands (giving the notion of the so-called "reduced basis" of a Hausdorff
topological vector space as a counterpart to the commonly used algebraic
basis of a vector space (Furthermore, the reduced basis is something fundamentally
different than a topological basis of a vector space.)). The eigenvalues
are as to be expected:

This
constitutes a new method of solving the energy-eigenproblem of the harmonic
oscillator in quantum theory.

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