# Tannor Quantum Mechanics derivative of variance of position

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1. Jan 16, 2016

### Quantum_of_Solace

0http://stackoverflow.com/questions/34833391/tannor-quantum-mechanics-derivative-of-variance-of-position# [Broken]

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/(m2)) * ( <p2> - <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/(m2))(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 <p2>?

Last edited by a moderator: May 7, 2017
2. Jan 21, 2016