Alternate approach to proving the virial theorem?

[chuckschuldiner]

Hi guys,
In quantum mechanics, the virial theorem for a system in its ground state is proved by a very nice scaling technique (Nielsen and Martin, PRB, 1985). I was trying to do something similar in classical mechanics and arrived at the virial theorem but i am not sure about why it should work.


Typically, the virial theorem (in 1-D) is proven as follows. Consider a set of interacting point masses Pi (i=1,2,3…N). The motion of Pi in an inertial frame is governed by

$$\[ f_i = \frac{d}{{dt}}\left( {m_i v_i } \right) \] \[\Rightarrow\ \sum\limits_i {x_i f_i } = \sum\limits_i {x_i \frac{d}{{dt}}\left( {m_i v_i } \right)} \] now since, \[ \sum\limits_i {x_i \frac{d}{{dt}}\left( {m_i v_i } \right)} = \frac{d}{{dt}}\left( {\sum\limits_i {x_i \left( {m_i v_i } \right)} } \right) - \sum\limits_i {v_i \left( {m_i v_i } \right)} \] we can write \[ \sum\limits_i {m_i v_i^2 } + \sum\limits_i {x_i f_i } = \frac{d}{{dt}}\left( {\sum\limits_i {m_i x_i v_i } } \right) \] Further if we assume that the position-momentum product on the right side of the above equation remains bound in time, then by taking a sufficiently long time average we can say \[ \left\langle {\sum\limits_i {m_i v_i^2 } } \right\rangle + \left\langle {\sum\limits_i {x_i f_i } } \right\rangle = 0 \]$$

where <> denotes the time-average.
This is the virial theorem.

Now, Please take a look at my approach. Please note that though virial theorem holds even if the system is not in equilibrium, i will consider a system of particles in its minimum energy configuration.

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The Hamiltonian for the n-particle system is given by

$$H= \[ \sum\limits_i^N {\frac{{p_i^2 }}{{2m_i }}} + V\left( {x_1 ,x_2 ,...,x_N } \right) \]$$

Now i will do a canonical transformation such that
$$\[ \begin{array}{l} x'_i \to \left( {1 + \upsilon } \right)x_i \\ p'_i \to p_i /\left( {1 + \upsilon } \right) \\ \end{array} \]$$

I know that i am just scaling the coordinates, but let us say i put these new coordinates in the expression for the Hamiltonian. Then, i will further say that since i started with a minimum energy configuration, the derivative of the energy w.r.t the parameter $$\upsilon$$ should be zero. Hence i should have,

$$\[ \frac{{\partial \left( {\sum\limits_i^N {\frac{{p_i^2 }}{{2m_i \left( {1 + \upsilon } \right)^2 }}} + V\left( {\left( {1 + \upsilon } \right)x_1 ,\left( {1 + \upsilon } \right)x_2 ,...,\left( {1 + \upsilon } \right)x_N } \right)} \right)}}{{\partial \upsilon }} = 0 \]$$

On simplification this gives the virial theorem back.

$$\[ \left\langle {\sum\limits_i {m_i v_i^2 } } \right\rangle + \left\langle {\sum\limits_i {x_i f_i } } \right\rangle = 0 \]$$

Why is this happening? What is the significance of the canonical transformation? And how come, i don't have to take any time-averaging? Is this calculation wrong?

I am a mechanical engineer..so please excuse me if there are any errors!

 

[luben]

hi schuldiner, just want to share some thoughts with you. i am not very familiar with the virial theorm, but i know it takes a meaning of averaging the spatial coordinates for a system. so it seems natural that when you scales the xi axes, they are averaged out eventially and any scaling of them does not matters. time is not averaged due to its definition (averaging in space is the only thing virial theorem concern, if i remember correctly). for the velocity space, since those derivatives are gone, it also does not participate in the final results. But for non-minimum E configuration, i think stuffs will be different. sorry i have not try to go thru those math steps by myself. that is interesting, though

thanks for the Q. just curious, are there any reason why you like to transform in that way?

 

[charng]

Dear fellow scientist, thank you for sharing your alternate approach to proving the virial theorem. It is always interesting to see different perspectives and methods in tackling scientific problems.

Firstly, let me commend you on your effort in trying to apply a similar scaling technique used in quantum mechanics to classical mechanics. This shows creativity and a deep understanding of the underlying principles.

In regards to your question about the significance of the canonical transformation, it is a powerful tool in classical mechanics that allows us to transform a given Hamiltonian into a simpler form, making it easier to solve. In your case, the transformation you applied is known as a 'scaling transformation' and it is commonly used in classical mechanics to simplify equations and reveal hidden symmetries.

Furthermore, the fact that you were able to arrive at the virial theorem by using this transformation is not a coincidence. The virial theorem is a special case of a more general theorem known as the 'Hamilton-Jacobi theorem', which states that for a system in equilibrium, the time-averaged kinetic energy is equal to the negative of the time-averaged potential energy. This result is independent of the specific form of the Hamiltonian and can be derived using different methods, including the one you have presented.

As for your question about the time-averaging, the reason why you do not have to explicitly take a time-average in your approach is because you are implicitly assuming that the system is in equilibrium. In classical mechanics, equilibrium is defined as a state where the system's macroscopic properties do not change with time. Therefore, by starting with a minimum energy configuration and using the Hamilton-Jacobi theorem, you are essentially assuming that the system is in equilibrium and hence, the time-averaging is not needed.

In conclusion, your approach to proving the virial theorem is valid and provides a different perspective on the underlying principles behind this important theorem in classical mechanics. Keep exploring and thinking outside the box, as it is through these alternate approaches that we can truly deepen our understanding of the natural world.