Completeness Relation: Significance & Multiplying State Vector

[wasi-uz-zaman]

hi, in quantum physics completeness relation is often use it equals to one - what is its significance in multiplying with state vector . thanks
wasi

 

[jtbell]

Do you mean this:

$$\sum {| n \rangle \langle n |} = 1\\
\sum {| n \rangle \langle n | \psi \rangle} = | \psi \rangle$$

This represents the expansion of $|\psi\rangle$ into a linear combination of orthonormal basis states $|n\rangle$. The $\langle n | \psi \rangle$ are the coefficients of the expansion. In the position representation we usually write this as something like

$$\psi(x) = \sum {c_n \psi_n(x)}$$