- #1
Denver Dang
- 143
- 1
Homework Statement
A particle is moving along the x-axis in the potential:
\[V\left( x \right)=k{{x}^{n}},\]
where k is a constant, and n is a positive even integer. \left| \psi \right\rangle is described as a normed eigenfunction for the Hamiltonoperator with eigenvalue E.
Show through the "Virial Theorem" that:
\[\begin{align}<br /> & \left\langle \psi \right|\hat{V}\left| \psi \right\rangle =\frac{2}{n+2}E \\ <br /> & \left\langle \psi \right|\hat{T}\left| \psi \right\rangle =\frac{2}{n+2}E, <br /> \end{align}\]<br />
where \hat{V}\ and \hat{T}\ denotes the operators respectively for potential and kinetic energy.
Homework Equations
The Virial Theorem:
\[2\left\langle T \right\rangle =\left\langle x\frac{dV}{dx} \right\rangle \]
The Attempt at a Solution
Well, I'm kinda lost.
I'm not sure how to calculate anything tbh...
The thing that confuses me, which is what I think I should do, is calculating:
\[\begin{align}<br /> & \left\langle \psi \right|\hat{V}\left| \psi \right\rangle \\ <br /> & \left\langle \psi \right|\hat{T}\left| \psi \right\rangle \\ <br /> \end{align}\]<br />
But can't find anything in my book that shows how to calculate anything that looks like that.
So a hint would be very helpful :)Regards