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[NOTES/QM-10003] A Summary of Coordinate and Momentum Representation

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A tabular summary of  coordinate and momentum representations is presented.

AK-47's picture 24-04-08 16:04:31 n

[NOTES/SM-04020] Gibbs Paradox

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kapoor's picture 24-04-08 10:04:31 n

[NOTES/ME-06005] Bounded Motion --- Oscillations Around Minimum

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AK-47's picture 24-04-08 07:04:01 y

[NOTES/ME-06002a]-General Properties of Motion in One Dimension

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AK-47's picture 24-04-08 07:04:13 y

[NOTES/ME-08003]-Effect of Earth's Rotation --Centrifugal Force

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AK-47's picture 24-04-08 07:04:57 n

[NOTES/CM-05005] Runge Lenz Vector

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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 \(-k/r\). 

kapoor's picture 24-04-08 04:04:16 n

[NOTES/SM-06001] Equilibrium Conditions for Open Systems

Node id: 6174page
kapoor's picture 24-04-07 16:04:13 n

[NOTES/CM-02007] Conservation of Energy

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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.

kapoor's picture 24-04-07 15:04:35 n

[NOTES/SM-06002] Gibbs Distribution

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kapoor's picture 24-04-07 13:04:11 n

[NOTES/SM-06004] Grand Canonical Ensemble --- Pressure and Chemical Potential

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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.

kapoor's picture 24-04-07 12:04:54 n

[NOTES/SM-06005] Entropy and Energy in Terms of Grand Canonical Partition Function

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In this article various thermodynamic functions are expressed in terms of the grand canonical partition function.

kapoor's picture 24-04-07 12:04:13 n

[NOTES/SM-06006] Mean Occupation Number for Identical Fermions and Bosons

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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.

kapoor's picture 24-04-07 12:04:52 n

[NOTES/SM-06003] Grand Canonical Ensemble

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In this article various thermodynamic functions are expressed in terms of the grand canonical partition function.

kapoor's picture 24-04-07 12:04:27 n

[NOTES/SM-04018] Boltzmann Entropy from Canonical Partition Function

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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.

kapoor's picture 24-04-06 19:04:11 n

[NOTES/SM-04019] Properties of an Ideal Gas

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Several properties such as internal energy, entropy etc. of perfect gases are calculated using the canonical partition function.

kapoor's picture 24-04-06 17:04:28 n

[NOTES/SM-04009] The Imperfect Gas

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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.

kapoor's picture 24-04-05 23:04:46 n

[NOTES/SM-04015] Equilibrium of a System with a Heat Reservoir

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For a system in equilibrium with a heat bath at temperature \(T\) the energy is not constant. We start with the Boltzmann relation \(S=k\log \Omega\). It will be shown that if of a microstate has energy \(E_r\), its probability \(p_r\) is proportional to \(e^{-\beta E_r}\) and is given by
\[p_r = \frac{\exp(-\beta E_r)}{Z} \]
where \(Z\) is function of \(T,V,N\), called the canonical partition function and is given by
\[Z = \sum_{\text{MS }r} \exp(-\beta E_r).\]

kapoor's picture 24-04-05 13:04:19 n

[NOTES/SM-04005] Applications of Canonical Ensemble to Paramanetism

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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.

kapoor's picture 24-04-05 13:04:43 n

[NOTES/SM-04010] Classical Theory of Specific Heat of Gases

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In this lecture we derive law of equipartition of energy under the assumption that the energy is quadratic function of some variable such as coordinates and momenta. As an application, the classical theory of specific heat of gases is given.

kapoor's picture 24-04-05 07:04:53 n

[NOTES/SM-04011] Specific Heat of Diatomic Gases --- Quantum Effects

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The quantum effect of  vibrations on specific heat of diatomic gases is presented. The results are in agreement with experimentally observed facts.

kapoor's picture 24-04-05 07:04:53 n

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