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In the Tannor textbook Introduction to Quantum Mechanics, there is a second derivative of chi on p37. It looks like this:

χ"(t) = d/dt ( (1/m) * (<qp + pq> - 2<p><q> ) (Equation 1)

Where χ is the variance of position, or χ= <q^2> - <q>^2 and q is position and p is momentum, and m is mass. Eventually this equation transforms into the following:

χ"(t) = (2/(m^{2})) * ( <p^{2}> - <p>^{2}- m*V"(<q>)*χ) (Equation 2)

There is obviously Ehrenfest theorem used here. I found that the first term inside the bracket in Equation 1 goes to 0 using integrals. The second becomes the following:

(2/m)*d/dt(<p><q>)

= (2/(m^{2}))(m*<q>V'(<q>)-m*(<q>*q-<q>^{2})V"(<q>) - <p>^{2})

and now I'm stuck. What's the next step? How do I get <p^{2}>?

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# Tannor Quantum Mechanics derivative of variance of position

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