- #1
bspoka
- 2
- 0
Hey does anybody have an idea of how to prove that
[tex]\frac{d}{dt}\left\langle{XP}\right\rangle= 2\left\langle{T}\right\rangle-\left\langle{x\frac{dV}{dx}}\right\rangle[/tex] for a hamiltonian of form
[tex]H=\frac{P^2}{2M}+V(x)[/tex]
where X is the position operator, P is the momentum and T is the kinetic energy. I got through the first chain rule and got
[tex]\frac{d}{dt}\left\langle{XP}\right\rangle=2\frac{\hbar}{i}\int{x\psi^*\frac{\partial^2\psi}{\partial t\partial x}}-\frac{\hbar}{i}\int{x\frac{\partial\psi^*}{\partial t}\frac{\partial\psi}{\partial x}[/tex]
with negative infinity to positive infinity bounds on the integrals. I tried integrating the first one by parts and it just gets even more messy.
I don't know where to go from here it's kind of starting to look like it should but I have no luck when I start substituting [tex]\frac{\partial\psi^*}{\partial t}[/tex] and [tex]\frac{\partial\psi}{\partial t}[/tex] from shcrodingers. Can anybode help?
[tex]\frac{d}{dt}\left\langle{XP}\right\rangle= 2\left\langle{T}\right\rangle-\left\langle{x\frac{dV}{dx}}\right\rangle[/tex] for a hamiltonian of form
[tex]H=\frac{P^2}{2M}+V(x)[/tex]
where X is the position operator, P is the momentum and T is the kinetic energy. I got through the first chain rule and got
[tex]\frac{d}{dt}\left\langle{XP}\right\rangle=2\frac{\hbar}{i}\int{x\psi^*\frac{\partial^2\psi}{\partial t\partial x}}-\frac{\hbar}{i}\int{x\frac{\partial\psi^*}{\partial t}\frac{\partial\psi}{\partial x}[/tex]
with negative infinity to positive infinity bounds on the integrals. I tried integrating the first one by parts and it just gets even more messy.
I don't know where to go from here it's kind of starting to look like it should but I have no luck when I start substituting [tex]\frac{\partial\psi^*}{\partial t}[/tex] and [tex]\frac{\partial\psi}{\partial t}[/tex] from shcrodingers. Can anybode help?