Compatibility

Compatibility is a property of two Operator.

If ∣ϕn⟩\left\lvert \phi_n \right\rangle is the nn-th eigenstate of A^\hat{A} (i.e., the Eigenbasis of A^\hat{A}),

Let

A^=∑n=1Nan∣ϕn⟩⟨ϕn∣\hat{A}=\sum_{n=1}^N a_n\left\lvert \phi_n \right\rangle\left\langle \phi_n \right\rvert

If B^\hat{B} is compatible with A^\hat{A} (i.e., B^\hat{B} is diagonalizable with A^\hat{A}) then,

B^=∑n=1Nbn∣ϕn⟩⟨ϕn∣\hat{B}=\sum_{n=1}^N b_n\left\lvert \phi_n \right\rangle\left\langle \phi_n \right\rvert