# 1.3: Operators and Observables


In addition to position, a full description of a system must contain some implicit information. The abstract bra-ket notation includes this.

Consider the electric charge. Obviously this is measurable, so it should be associated with an operator $$\hat{Q}$$, such that e.g.

$\hat{Q}|\Phi \rangle = −e| \Phi \rangle\nonumber$

where $$\Phi$$ is the wavefunction of an electron. $$−e$$ meets all the criteria for a quantum number, and the above equation is obviously a true representation of reality. Thus the meaning of the ket $$|\Phi \rangle$$ is broader than a simple spatial function, and operators can also be non-algebraic. This is especially important in particle physics where all manner of quantum numbers appear (isospin, strangeness, baryon number etc. etc.)

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