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The steps: $$<x> = \int_{-\infty}^{\infty} \psi^*(x,t)x\psi(x,t)dx$$ $$= (2 \pi h)^{-1} \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} exp [-i (p'_x x - p_x^{'2} t / 2 m)/h] \phi^* (p'_x) dp'_x (ih \frac{\partial}{\partial p_x})$$ $$\int_{-\infty}^{\infty} exp [i (p_x x - p_x^2 t / 2 m)/h]...

Hi, I am wondering the extra term I get, which is different from what is asked to prove in the hints, ie, $$\int_{-\infty}^{\infty} -\phi^*(p_x) x \phi(p_x) dp_x$$ is equal to zero?

Homework Statement Show that, for a general one-dimensional free-particle wave packet $$\psi (x,t) = (2 \pi h)^{-1/2} \int_{-\infty}^{\infty} exp [i (p_x x - p_x^2 t / 2 m)/h] \phi (p_x) dp_x$$ the expectation value <x> of the position coordinate satisfies the equation $$<x> = <x>_{t=t_0}...

I know how to prove it now. Thanks for the help.

I've tried integration, and get $\frac{\arctan(\frac{x}{\epsilon})}{\pi}$. However, seems there is still no way out.

Hi, I am reading the Quantum Mechanics, 2nd edition by Bransden and Joachain. On page 777, the book gives an example of Dirac delta function. $\delta_\epsilon (x) = \frac{\epsilon}{\pi(x^2 + \epsilon^2)}$ I am wondering how I can show $\lim_{x\to 0+} \int_{a}^{b} \delta_\epsilon (x) dx$...