Completeness in Quantum Mechanics While studying Matrix representation I found a topic about completeness for a basis by the equation:$
∑_{n}|ψ_n⟩⟨ψ_n|=1$
I don't understand the physical meaning behind it. Is it something to do with infinite basis vectors?
 A: If you recall the projection operator, $\mathcal{P}$, which look like
$$\mathcal{P}=|\phi_n\rangle \langle \phi_n|$$
where $\{|\phi_i\rangle\}$ are assumed to be the basis set for the LVS. Then what this operator does is project a component of any arbitrary vector along with the basis $|\phi_n\rangle$.
What I'm trying to say, Given a vector
$$|\psi\rangle =\sum_i c_i|\phi_i\rangle $$
$$\mathcal{P}|\psi\rangle =\sum_ic_i|\phi_n\rangle \langle \phi_n|\phi_i\rangle=c_n|\phi_n\rangle  $$
That explain the projection operator.
Now If I project the along all its component,
$$\sum_i \mathcal{P_i}|\psi\rangle =\sum_i|\phi_i\rangle \langle\phi_i|\left(\sum_jc_j|\phi_j\rangle\right)=\sum_{i,j}c_j|\phi_i\rangle \langle \phi_i|\phi_j\rangle =|\psi\rangle $$
In other world,
$$\sum_i|\phi_i\rangle \langle \phi_i|=I$$
This is saying nothing but If I project the vector along all its basis vectors, I get the vector back.
A: This is just the statement that any state can be written as a linear combination of the elements in the basis.
