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I wonder what the name of this normalization process is

  1. Jul 12, 2015 #1

    td21

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    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.
     
    Last edited: Jul 12, 2015
  2. jcsd
  3. Jul 13, 2015 #2
    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.
     
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