How to Calculate the Virial Theorem for a Particle Moving in a Potential?

[Denver Dang]

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:

$$\[\begin{align} & \left\langle \psi \right|\hat{V}\left| \psi \right\rangle =\frac{2}{n+2}E \\ & \left\langle \psi \right|\hat{T}\left| \psi \right\rangle =\frac{2}{n+2}E, \end{align}\]$$
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:

$$\[\begin{align} & \left\langle \psi \right|\hat{V}\left| \psi \right\rangle \\ & \left\langle \psi \right|\hat{T}\left| \psi \right\rangle \\ \end{align}\]$$

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

 

[George Jones]

Start with

$$\left\langle \psi \right|\hat{H}\left| \psi \right\rangle = \left\langle \psi \right|\hat{T} + \hat{V} \left| \psi \right\rangle.$$

What is the left side? What is the right side?

 

[Denver Dang]

Sorry for the late reply...

But that is my problem. I'm not sure how to calculate that ?
Is it an integral, a commutator trick, or...? As I said, I can't seem to find anything in my book that shows how to calculate that.

 

[George Jones]

Start at the beginning. What is

$$\left\langle \psi \right|\hat{H}\left| \psi \right\rangle?$$

 

[Cyosis]

Hint to George Jones' question:

$\left| \psi \right\rangle$ is described as a normed eigenfunction for the Hamiltonoperator with eigenvalue E.