Definition of outer product

Question

I am trying to understand the concept of outer product in quantum mechanics. I read "Quantum Computing explained" of David MacMahon.

I can understand the transition in (3.12): $$(|\psi\rangle \langle \phi | )|\chi\rangle \rightarrow |\psi\rangle \langle \phi |\chi\rangle $$

But how to get $(\langle \phi | \phi | \chi \rangle ) | \psi \rangle$ ?

Why it is possible to get through such steps?

1. $|\psi\rangle \langle \phi | \chi\rangle $
2. $|\psi\rangle \langle \phi | \phi | \chi\rangle $
3. $\langle \phi | \phi | \chi\rangle |\psi\rangle $

Answer 1

I'm thinking that it's a typo, and all the author wanted in the last term was to write $$ (\langle \phi|\chi\rangle)\,|\psi\rangle. $$ The proof uses that you have a kind of associativity in the first equality $(|\psi\rangle\langle\phi|)\,|\chi\rangle= |\psi\rangle\,\langle\phi|\chi\rangle$ which I think is brought out of the blue if you introduce bras and kets out of nowhere.

The equality is obvious if you notice that kets as simple column vectors in $\mathbb C^n$, and bras are their adjoints (conjugate transpose). In that setting your equality is $$ (\psi\phi^*)\,\chi=\psi\,(\phi^*\chi)=(\phi^*\chi)\,\psi, $$ where the associativity is that of the product of matrices.