I wonder what the name of this normalization process is

[td21]

I wonder what the name of this normalization process is for better reference.

The scenario is like this:

$$\left|\Psi\right> = \frac{1}{\sqrt{6}}\left(\left|a\right>+\left|b\right>+\left|c\right>+\left|d\right>+\left|e\right>+\left|f\right>\right)$$
where each of the components inside the bracket is orthonormal to each other.

$$M$$ is an operator which is non-hermitian.
$$M\left|\Psi\right> = \frac{1}{\sqrt{6}}\left(\left|a'\right>+\left|b'\right>+\left|c'\right>+\left|d'\right>+\left|e'\right>+\left|f'\right>\right).$$

If $$\left|a'\right>=\left|d'\right>$$, $$\left|b'\right> = \left|e'\right>$$, $$\left|c'\right> = \left|f'\right>$$,

then $$H\left|\Psi\right> = \frac{2}{\sqrt{6}}\left(\left|a'\right>+\left|b'\right>+\left|c'\right>\right).$$

We have to normalize this new state. What is this normalization principle called in quantum mechanics or any textbook regarding this? Thank you very much.

 

[Cadaei]

I don't know if it has a name but it simply comes from your non-hermitian operator, which as I understand it are only used as a mathematical convenience in quantum mechanics. What matters is that they have real eigenvalues so they can correspond to observables. The choice of normalization constant was arbitrary to begin with, since A*psi is a solution if psi is one, I don't think it matters very much... you can just accept it as a consequence of having non-orthogonal eigenvectors as a result of non-hermiticity and renormalize.