Use the Virial theorem to show the following...

[GravityX]

Homework Statement

I have to show the following with the virial theorem $E=\langle \cal H \rangle_k$ $=k_bT \sum\limits_{i=1}^{N}a_i$

Relevant Equations

none

The expression $\langle \cal H \rangle_k$ is the expected value of the canonical ensemble.

The Hamiltonian is defined as follows, with the scaling $\lambda$

$\lambda \cal H$ : $\lambda H(x_1, ...,x_N)=H(\lambda^{a_1}x_1,....,\lambda^{a_N}x_N)$

As a hint, I should differentiate the Hamiltonian with respect to $\lambda$

Unfortunately, I don't know how exactly I should do this, i.e. form the derivative, I have now proceeded in such a way that I have formed the differential

$$H(\lambda^{a_1}x_1,...\lambda^{a_N}x_N)=\lambda^{a_1}x_1\frac{\partial H}{\partial x_1}+...\lambda^{a_N}x_N\frac{\partial H}{\partial x_N}$$

Then the derivative should look like this,

$$\frac{\partial}{\partial \lambda}H(\lambda^{a_1}x_1,...\lambda^{a_N}x_N)=a_1\lambda^{a_1-1}x_1\frac{\partial H}{\partial x_1}+...a_N\lambda^{a_N-1}x_N\frac{\partial H}{\partial x_N}$$

Is this correct

 

[vanhees71]

Not entirely. Just take the total derivative of the equation
$$\lambda H(x_1,\ldots,x_N)=H(\lambda^{a_1} x_1,\ldots,\lambda^{a_N} x_N).$$
Your last equation is correct.

Next you have to think about, what
$$\left \langle x_k \frac{\partial H}{\partial x_k} \right \rangle$$
might be. For that start with the definition of how to take expectation values in the canonical ensemble. It's not too easy, and maybe you are allowed to use the result, which is the "virial theorem".

 

[GravityX]

Thank you vanhees71 for your help and sorry I'm only getting back to you now, I had two weeks Christmas break

I was able to solve the problem now, the expression $\Bigl\langle x_k\frac{\partial H }{\partial x_k} \Bigr\rangle$ we had stated in the lecture as $k_bT$.