Fermi Golden Rule says:

$\Gamma \propto |M_{ij}|^2$

I know how to get $M_{ij}$, but how do I proceed? How do I take a norm of Hermitian matrix? There is no clear (to me) definition in the internet except the Frobenius norm, and I don't think that this is it.

I have assumed all my life that

$|M_{ij}|^2 = M_{ij}^{\dagger}M_{ij}$

but since I have got a complex number where I have not expected it, I am not sure any more.

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    $\begingroup$ If you got a complex number there, then you likely made a mistake in your calculation. $\endgroup$ – probably_someone Feb 9 '18 at 21:47

The notation just means the magnitude of the complex number $M_{ij}$ for $i$ and $j$ labelling the initial and final states. $$ |M_{ij}|^2= |\langle i \vert M\vert j \rangle|^2. $$ It's not intended to be any of the matrix norms of the operator $M$.

  • $\begingroup$ $M_{ij}$ is a complex matrix (!), not a complex number; it's not an expectation value, it is transition MATRIX. If I proceed, I get $\langle j|\hat{M}^{\dagger}|i\rangle \langle i|\hat{M} |j\rangle = \langle j|\hat{M}^{\dagger}\hat{M}|j\rangle$, and from here I am stuck. Also, is it $|M_{ij}|^2 = \langle j|\hat{M}^{\dagger}|i\rangle \langle i|\hat{M} |j\rangle$ or $|\langle j|\hat{M}^{\dagger}|i\rangle \langle i|\hat{M} |j\rangle|$? Not clear... $\endgroup$ – MsTais Feb 9 '18 at 22:11
  • $\begingroup$ I have never really thought about that, but def of norm of a vector in Hilbert space does not translate into norm of a matrix automatically, so construction $|M_{ij}|^2$ is not that obvious, apparently... $\endgroup$ – MsTais Feb 9 '18 at 22:13
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    $\begingroup$ @MsTais I always thought it was a matrix element. There is but one $i$, the initial state. The final state is usually a plane wave with some density of states in phase space--so you only calculate the magnitude of the matrix element from the initial state to the final state. $\endgroup$ – JEB Feb 9 '18 at 22:18
  • $\begingroup$ Uf, yes, in ideal world you are given initial and final states and you can collapse everything to numbers... In general it is a matrix, a mapping, you supply $|i\rangle$ and $|f\rangle$ of the system and get the probability of this transition to happen. I can't let it collapse to a number. I need to keep its general form. $\endgroup$ – MsTais Feb 9 '18 at 22:36

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