# Expressing a quantum mechanical state as a linear combination of the basis kets of an observable

I have a question. Suppose we have a quantum mechanical system, of which we can measure 3 different observables. Each of the observables has an eigenvector spectrum and we can express a state as a linear combination of the eigenvectors (we assume that they form a basis). My question is the following.

The system is in particular state $$|\psi\rangle$$.
If we want to express it as a linear combination of basis vectors, which basis vectors do we use (we have 3 different types since we have 3 observable)?

Or maybe, if we assume that the observables commute and form a complete set, is the basis made of eigenvectors that belong to all 3 observables?

Now let's say we have an observable that has a discrete basis and another that has a continuous one. Assuming that they commute, will the basis with eigenvectors of both of them be a mix of a discrete and a continuous eigenvector spectrum?

All representations of the state are equivalent in Quantum mechanics. For a state $$|\alpha⟩$$ with two observables $$A$$ and $$B$$ over the space $$\{|a'⟩\}$$ and $$\{|b'⟩\}$$, it's possible to write $$|\alpha⟩= \sum{⟨a'|\alpha⟩|a'⟩}$$ $$|\alpha⟩= \sum{⟨b'|\alpha⟩|b'⟩}$$ Both of these representations are equivalent. This can be seen by considering a unitary transformation, $$|b'⟩= U|a'⟩$$ Without much effort it can be seen that $$U$$ has the form, $$U=\sum |b'⟩⟨a'|$$ Substituting this transformation, $$|\alpha⟩= \sum{⟨b'|\alpha⟩|b'⟩} = \sum{⟨a'|U^{\dagger}|\alpha⟩U|a'⟩}$$ Hence the expansion coefficients of $$|\alpha⟩$$ in $$b'$$ representation is $$⟨a'|U^{\dagger}|\alpha⟩$$. Also the operator $$A$$ in $$b'$$ representation will become $$U^{\dagger}AU$$. Similar procedure is to be followed for continuous basis. This is irrespective of whether the operators commute or not. If the operators commute then a common set of eigen basis exists for both operators. In the case of compatible observables, the basis are generally written as $$|a',b'⟩$$.