Tannor Quantum Mechanics derivative of variance of position

[Quantum_of_Solace]

0http://stackoverflow.com/questions/34833391/tannor-quantum-mechanics-derivative-of-variance-of-position# 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>?

 

[eljose79]

Hello,

Thank you for your question. It seems like you are working on the second derivative of the variance of position in quantum mechanics. To get the <p^2> term, you can use the commutation relation between the position and momentum operators, [p,q]=iħ, and the definition of the momentum operator, p= -iħ(d/dq). This will allow you to express <p^2> in terms of the position operator and its derivatives. You can then use integration by parts to evaluate the integral and simplify the expression. I hope this helps. Good luck!