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given I know the density matrix elements in position basis as a function of time.

$ \langle x | \rho(t) |x' \rangle$

How do I calculate the expection values like $\langle x^2(t) \rangle $ or $ \langle p^2(t) \rangle$ from this?

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The expectation values are calculate with this formula

$$ \langle \hat A \rangle = Tr(\hat A \rho) = Tr(\rho \hat A). $$

Since the trace is basis-independent you can evaluate the trace in an arbitrary basis $\{ |n\rangle \}$

$$ Tr(\hat A \rho) = \sum_n \langle n | \hat A \rho | n \rangle = \sum_n \langle n | \hat A \sum_m | m \rangle \langle m |\rho | n \rangle = \sum_{n,m} \langle n | \hat A | m \rangle \langle m | \rho | n \rangle, $$

where we have used a completeness relation $\sum_m |m \rangle \langle m|$. Now you can simply use the position basis instead of the arbitrary basis $\{ |n\rangle \}$.

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I think I got it myself:

$\rho = \int dx dx' \langle x |\rho| x' \rangle |x \rangle \langle x' |$

Then it follows:

$\langle x^2 \rangle = \int dx \langle x |\rho| x \rangle x^2$,

right?

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  • $\begingroup$ Yes, looks good! $\endgroup$ – MST Nov 26 '19 at 13:39

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