Browse & Filter

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It is proved that the Runge-Lenz vector \begin{equation}
\vec{N} = \vec{v} \times \vec{L} - \frac{k\vec{r}}{r}
\end{equation} is a constant of motion for the Coulomb potential \(-k/r\). 

If the Lagrangian does not depend on time explicitly, the Hamiltonian \(H=\sum_{k=1}\frac {\partial{L}}{\partial{\dot q_k}}\dot q_k-L\) is a constant of motion. For conservative systems of many particles, the Hamiltonian coincides with the total energy. is conserved.

Using thermodynamics to express TdS in terms of Gibbs energy and comparing TdS expression in terms of the grand canonical partition function we obtain pressure and chemical potential function in terms of grand canonical partition function.

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Using occupation number representation for identical particles, the grand canonical partition function is expressed in terms of single particle partition function. This is then used to discuss cases of identical fermions and identical bosons. Mean occupation number for fermions and bosons is obtained.

We derive the Boltzmann relation for entropy using the canonical partition function. This derivation uses the fact that the number of micro states \(\Omega(E)\) as a function energy has a sharp peak around the mean energy.

The ideal gas equation \(pV=NkT\) is good approximation for low densities. In this section a scheme of obtaining corrections to the ideal gas equation is discussed.

In this section the method of canonical ensemble is applied to paramagnetism. For a paramagnetic substance the variation of paramagnetic susceptibility with temperature is derived.