# 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

## 2 Answers

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.

If you are in a non-orthogonal basis the projections are not unique.

If it is not a normalized basis you must normalize the coefficients to get the correct probabilities.

You should always try to work in a orthonormal basis.

It's convention to normalize the wave function, but this is purely something done to make calculations easier. States multiplied by a c number(complex number) are completely equivalent.

This is why we choose to normalize states (magnitude 1). It makes the calculations for things like the transitions probability between states easier. If you include your constant your just have to divide by the norm anyways in this kind of calculation, which is equivalent to having a normalized state to begin with.