Exploring the Physical Significance of the Virial Theorem

[maverick280857]

Hi everyone

I have a question regarding a step in the proof of the Virial Theorem.

Specifically suppose $|E\rangle$ is a stationary state with energy $E$, i.e.

$$\hat{H}|E\rangle = E|E\rangle$$

Now,

$$[\hat{r}\bullet\hat{p},\hat{H}] = i\hbar\left(\frac{p^2}{m} - \vec{r}\bullet\nabla V\right)$$

Taking the expectation value of the left hand side over stationary states, we see that

$$\langle E|[\hat{r}\bullet\hat{p},\hat{H}]|E\rangle = 0$$

(The Virial Theorem for central potentials then assumes $V(r) = \alpha r^{n}$ and one gets <T> = (n/2)<V>.)

My question is: what is the physical significance of this commutator and what does it mean physically that the expectation of this commutator wrt a basis of stationary states is zero?

Thanks in advance.

Cheers,
Vivek.

 

[maverick280857]

Anyone?

 

[Entropia]

I would like to start by commending you on your thorough understanding of the Virial Theorem and your insightful question. The Virial Theorem is an important concept in physics, specifically in the study of systems that have a potential energy that depends only on the distance between particles, known as central potentials. It relates the average kinetic energy of a system to the average potential energy, and is a useful tool for understanding the behavior of these systems.

To answer your question, let's first look at the physical significance of the commutator [\hat{r}\bullet\hat{p},\hat{H}]. This commutator represents the uncertainty in the position and momentum of a particle, as well as the effect of the potential energy on the system. In simpler terms, it tells us how the position and momentum of a particle change in response to the potential energy of the system.

Now, when we take the expectation value of this commutator over stationary states, we are essentially averaging out the effects of the potential energy on the system. This is because stationary states are eigenstates of the Hamiltonian, and thus the potential energy remains constant in these states. Therefore, the expectation value of the commutator becomes zero, indicating that the effects of the potential energy on the system are balanced out by the uncertainty in the position and momentum of the particles.

In the case of central potentials, where the potential energy is proportional to the distance between particles, the Virial Theorem tells us that the average kinetic energy is equal to half the average potential energy, multiplied by a factor depending on the form of the potential. This provides us with a useful relationship between the kinetic and potential energies of a system, allowing us to better understand its behavior.

I hope this helps to answer your question and further your understanding of the Virial Theorem. Keep exploring and questioning, as this is the essence of science. Best of luck in your studies.