A superposition of states such as [tex]a_1|\psi_1\rangle+...+a_n|\psi_n\rangle[/tex] represents a single physical state, a state for which the probability of a measurement finding the system in state [tex]|\psi_k\rangle[/tex] is [tex]|a_k|^2[/tex]. The [tex]a_k[/tex] represent "quantum-type" probabilities.(adsbygoogle = window.adsbygoogle || []).push({});

On the other hand the density matrix [tex]D=p_1|\psi_1\rangle\langle\psi_1|+...+p_n|\psi_n\rangle\langle\psi_n|[/tex] represents a statistical ensemble of states, statistical in the classical sense in which we accept that the systemreallyis in some particular state but we just don't know which one. The probabilities [tex]p_k[/tex] reflect the uncertainty in our knowledge of the system, the kind of ordinary probabilities that would apply to, say, a poker game.

I wrote the above just to verify that I understand it this far. Ok? (I know there are large ontological and epistemological gray areas in the above statements, but let's just go with the Copenhagen interpretation for the sake of discussion.)

Now if we switch to some other set of basis states [tex]|\phi_k\rangle[/tex], this same density matrix D will contain "off-diagonal" terms [tex]|\phi_j\rangle\langle\phi_k|, j\ne k[/tex]. If a term [tex]p_k|\psi_k\rangle\langle\psi_k|[/tex] means "[tex]p_k[/tex] is the probability that the system is actually in state [tex]|\psi_k\rangle[/tex]," then what do the terms of the form [tex]q_{jk}|\phi_j\rangle\langle\phi_k|, j\ne k[/tex] represent?

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# Density matrix off diagonal terms - what do they mean?

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