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## Main Question or Discussion Point

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

[tex]

\[

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

\]

[/tex]

where <> denotes the time-average.

This is the virial theorem.

-------------------------------------------------------------------------------------------------------------

The Hamiltonian for the n-particle system is given by

[tex]

H=

\[

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

\]

[/tex]

Now i will do a canonical transformation such that

[tex]

\[

\begin{array}{l}

x'_i \to \left( {1 + \upsilon } \right)x_i \\

p'_i \to p_i /\left( {1 + \upsilon } \right) \\

\end{array}

\]

[/tex]

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 [tex]\upsilon[/tex] should be zero. Hence i should have,

[tex]

\[

\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

\]

[/tex]

On simplification this gives the virial theorem back.

[tex]

\[

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

\]

[/tex]

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

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

[tex]

\[

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

\]

[/tex]

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.**-------------------------------------------------------------------------------------------------------------

The Hamiltonian for the n-particle system is given by

[tex]

H=

\[

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

\]

[/tex]

Now i will do a canonical transformation such that

[tex]

\[

\begin{array}{l}

x'_i \to \left( {1 + \upsilon } \right)x_i \\

p'_i \to p_i /\left( {1 + \upsilon } \right) \\

\end{array}

\]

[/tex]

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 [tex]\upsilon[/tex] should be zero. Hence i should have,

[tex]

\[

\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

\]

[/tex]

On simplification this gives the virial theorem back.

[tex]

\[

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

\]

[/tex]

Why is this happening? What is the significance of the canonical transformation? And how come, i dont have to take any time-averaging? Is this calculation wrong?Why is this happening? What is the significance of the canonical transformation? And how come, i dont have to take any time-averaging? Is this calculation wrong?

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