# Calculating the Wavefunction Associated with any Ket Vector

We can express any ket vector $$|\psi⟩$$ (representing all the different possible states) in its “component form” as $$|\psi⟩=\sum_{i=1}^{N}\psi_i|i⟩$$ where $$|i⟩$$ are all the different possible basis vectors one could decompose $$|\psi⟩$$ with respect to, $$\psi_i$$ are the different components (which, in general, can be complex numbers), and the value of $$N$$ is simply the number of dimensions in the space. The basis vectors $$|i⟩$$ are by definition orthogonal vectors whose magnitudes equal one. This is analogous to the unit vectors $$\hat{i}$$, $$\hat{j}$$, and $$\hat{k}$$ which are, by definition, perpendicular vectors whose magnitudes equal 1.

We shall now derive an equation which will allow us to find the components $$\psi_i$$ of any complex vector . Let’s start by taking the inner product between $$|\psi⟩$$ and any basis vector $$|j⟩$$ to get

$$⟨j|\psi⟩=\sum_{i=1}^{N}\psi_i⟨j|i⟩.$$

If $$i≠j$$, then we are considering the inner product between two different basis vectors which, by definition, are orthogonal. Therefore all the terms $$⟨j|i⟩$$ in the sum in which $$i≠j$$ are zero. When $$i=j$$, we are taking the inner product between the same two basis vectors which have equal magnitudes of one and point in the same direction; thus $$⟨j|i⟩$$=1\) when $$i=j$$. This means that the inner product $$⟨j|i⟩$$ simply is just the Kronecker delta $$𝛿_{ij}$$; thus $$⟨j|i⟩=𝛿_{ij}$$ and

$$⟨j|\psi⟩=\sum_{i=1}^{N}\psi_i𝛿_{ij}.$$

In the sum, all of the terms become zero except for the $$\psi_j𝛿_{jj}=α_j$$ term. Thus, the equation simplifies to

$$⟨j|\psi⟩=\psi_j.\tag{23}$$

This result indicates that we're at the halfway point towards our goal of deducing that $$\psi_i=⟨L_i|\psi⟩$$—something I said, earlier, that I'd eventually prove. The only discrepancy between this equation and Equation (23) is the dummy variable (which doesn't matter) and the fact that there's the bra $$⟨j|$$ instead of $$⟨L_i|$$. But once we prove that the eigenvectors $$|L_i⟩$$ of any observable $$\hat{L}$$ form a complete orthonormal basis, then you'll be able to breath a sigh of relief. Since this analysis applies to any set of basis vectors $$|i⟩$$, the generality of this analysis permits us to substitute $$|L_i⟩$$ for $$|i⟩$$ in order to obtain our desired result.