# Expansion of a ket-physical interpretation of coefficients

Consider I have a state represented by the Ket: $$|\psi\rangle=\sum_i a_i |\phi_i\rangle$$ What are the physical interpretations of the coefficients $a_i$? My guess is that $|a_k|^2$ represents the probability that the 'paritcle' is in the state $|\phi_k\rangle$ is this right? If so does it only hold for orthonormal basis or any basis?

• "My guess is that $|a_k|^2$ represents the probability that the 'particle' is in the state $|\phi_k\rangle$ is this right?" If the state is being described by the ket $|\psi\rangle$, it cannot be described also by $|\phi_k\rangle$ at the same time. One has to choose one. – Ján Lalinský May 11 '15 at 7:06

The coefficients $a_k$ quantity the projection of the state $|\psi\rangle$ onto the $k^\mathrm{th}$ basis state. So if you measure in that basis you would expect $|a_k|^2$ of the time to measure state $|\phi_k\rangle$. If you change the basis you will need to recalculate the projection coefficients.