# Linear operator and linear vector space?

## Main Question or Discussion Point

hi, please tell me what do we mean when we say in quantum mechanics operators are linear and also vector space is also linear ?

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bhobba
Mentor
Its to do with the axioms of QM.

At the starting level we say the states form a vector space - and by definition vector spaces are linear - as hopefully you have learnt in a course of linear algebra.

The first axioms of QM is that observables are hermitian linear operators such that the eigenvalues (necessarily real since the operator is hermitian) are the possible outcomes of the observation associated with the observable.

The second axiom is the so called Born rule. If a system is in state u and it is observed with observable O the expected value of the outcome is <u|O|u>.

There is more that can be said at an advanced level - especially the very important Gleason's Theorem - see post 137:

But basically linear is associated the vector space language QM is expressed in.

Thanks
Bill

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