Question

\[\frac{\text d \langle {p} \rangle}{ \text{d} t} \]

Answer 1

\[= \frac{\text d}{\text d t} \int\limits_{-\infty}^{\infty} \Psi_{(x,t)}^* \left( \frac{\hbar}{i}\frac{\partial}{\partial x} \right) \Psi_{(x,t)} ~\text d x\]
\[= \frac{\hbar}{i}\int\limits_{-\infty}^{\infty} \frac{\partial }{\partial t} \left(\Psi^*\frac{\partial\Psi}{\partial x}\right)~\text d x \]
\[= \frac{\hbar}{i}\int\limits_{-\infty}^{\infty} \frac{\partial \Psi^*}{\partial t}\frac{\partial \Psi}{\partial x} +\Psi^*\frac{\partial }{\partial x}\frac{\partial \Psi }{ \partial t} ~\text {d} x\]
\[= \frac{\hbar}{i}\int\limits_{-\infty}^{\infty}\left( -\frac{i\hbar}{2m}\frac{\partial^2 \Psi^*}{\partial x^2}+\frac{i}{\hbar} V\Psi^*\right)\frac{\partial \Psi}{\partial x}+\Psi^* \frac{\partial}{\partial x} \left(\frac{i\hbar}{2m} \frac{\partial^2 \Psi }{\partial x^2} -\frac{i}{\hbar}V\Psi \right)~\text d x\]
\[\vdots\]

Answer 2

use integration by parts twice to simplify

Answer 3

\[= \frac{\hbar}{i} \int\limits_{-\infty}^{\infty} \frac{i\hbar}{2m}\left( \Psi^* \frac{\partial^3 \Psi}{\partial x^3} -\frac {\partial^2 \Psi^*}{\partial x^2} \frac{\partial \Psi}{\partial x} \right)+ \frac{i}{\hbar} \left( V \Psi^* \frac{\partial \Psi}{\partial x}-\Psi^* \frac{\partial}{\partial x} (V \Psi) \right)~\text d x\]
\[\vdots \]%=\]
\[= \frac{\hbar}{i} \int\limits_{-\infty}^{\infty} \frac{i}{\hbar} \left( V \Psi^* \frac{\partial \Psi}{\partial x}-\Psi^*V \frac{\partial \Psi}{\partial x} - \Psi^* \frac{\partial V}{\partial x} \Psi \right)~\text d x\]
\[= \int\limits_{-\infty}^{\infty} -\Psi^* \frac{\partial V}{\partial x} \Psi ~\text {d} x\]
\[= \int\limits_{-\infty}^{\infty} - \frac{\partial V}{\partial x} |\Psi|^2~\text {d} x\]
\[=\left\langle - \frac{ \partial V }{\partial x} \right\rangle \]

Answer 4

can some one help me to apply integration by parts twice to this

\[= \frac{\hbar}{i}\int\limits_{-\infty}^{\infty}\left( -\frac{i\hbar}{2m}\frac{\partial^2 \Psi^*}{\partial x^2}+\frac{i}{\hbar} V\Psi^*\right)\frac{\partial \Psi}{\partial x}+\Psi^* \frac{\partial}{\partial x} \left(\frac{i\hbar}{2m} \frac{\partial^2 \Psi }{\partial x^2} -\frac{i}{\hbar}V\Psi \right)~\text d x\]
\[\vdots\]