(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

A particle is moving along the x-axis in the potential:

[tex]\[V\left( x \right)=k{{x}^{n}},\][/tex]

where [itex]k[/itex] is a constant, and [itex]n[/itex] is a positive even integer. [itex]\left| \psi \right\rangle [/itex] is described as a normed eigenfunction for the Hamiltonoperator with eigenvalueE.

Show through the "Virial Theorem" that:

[tex]\[\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}\]

[/tex]

where [itex]\hat{V}\[/itex] and [itex]\hat{T}\[/itex] denotes the operators respectively for potential and kinetic energy.

2. Relevant equations

The Virial Theorem:

[tex]\[2\left\langle T \right\rangle =\left\langle x\frac{dV}{dx} \right\rangle \][/tex]

3. 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:

[tex]\[\begin{align}

& \left\langle \psi \right|\hat{V}\left| \psi \right\rangle \\

& \left\langle \psi \right|\hat{T}\left| \psi \right\rangle \\

\end{align}\]

[/tex]

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

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# The Virial Theorem problem

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