I was reading Griffiths Introduction to Quantum Mechanics and I came across this equation: $$\langle \psi_a |z|\psi_b\rangle$$ Where z is direction. I don't understand why there are those subscripts a and b and what do we get from this bra-ket. I know that without subscripts it would be expected value of z, but with this notation I have no idea.
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2$\begingroup$ Could you indicate the name of the chapter? (edition and page would also help). $\endgroup$– MauricioSep 21, 2021 at 18:52
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$\begingroup$ @Mauricio 11.2.1 3rd edition, page 411 $\endgroup$– coverSep 21, 2021 at 18:53
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$\begingroup$ Instead of being an expected value, this is more of a general inner product or transition amplitude. If you know how to look at the overlap between two states, like the inner product between $|\Psi\rangle$ and $|\Phi\rangle$ being $\langle \Phi|\Psi\rangle$, then you'll be able to compute this: first determine the action of $z$ on $|\psi_b\rangle$ and then compute the overlap between the new "state" $z|\psi_b\rangle$ and $|\psi_a\rangle$. $\endgroup$– Quantum MechanicSep 21, 2021 at 19:00
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$\begingroup$ The point is that the two subscripts make them two different states. These calculations crop up all over the place in QM even if they don't directly have physical meaning (specifically, they tell could you how to express the elements of the operator $z$ in an orthonormal basis of which two elements are $|\psi_a\rangle$ and $|\psi_b\rangle$). $\endgroup$– Quantum MechanicSep 21, 2021 at 19:02
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$\begingroup$ Have you encountered the alternative notation $\langle a|\hat{z}|b\rangle$? $\endgroup$– J.G.Sep 21, 2021 at 19:03
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I do not have the 3rd edition but looking at the index I can see it corresponds to time-dependent perturbation theory. In general, $|\psi_a\rangle$ and $|\psi_b\rangle$ are two states. Imagine that you have some interaction that can take your initial state $|\psi_a\rangle$ into $|\psi_b\rangle$ and you wish to know the probability of that happening. Fermi's golden rule is an advanced formula that allows you to calculate the transition rate (the probability of that happening).
What you need to apply Fermi's golden rule is called an amplitude $\langle \psi_b|H_{\rm I}|\psi_a\rangle$ where $H_{\rm I}$ is the interaction Hamiltonian. For electromagnetic waves, this interaction Hamiltonian is associated to the electric dipole which is proportional to the distance in one direction of space (in the coordinates chosen in the book is $z$). So
$$\langle \psi_b|z|\psi_a\rangle$$
is a probability amplitude that measures the projection of $|\psi_b\rangle$ onto $|\psi_a\rangle$ (the overlap) after a given modification (interaction proportional to $z$) . More importantly,
$$|\langle \psi_b|z|\psi_a\rangle|^2$$ is proportional to the transition rate. This kind of amplitudes are very important in particle physics when calculating the results from scattering experiments.
