Discuss the concept of thermodynamic fluctuations in the canonical ensemble. Define energy fluctuations and derive an expression for the mean square fluctuation of energy. Using a classical ideal gas, show that the relative fluctuation in energy varies as and hence, justify why thermodynamic calculations are valid for ordinary macroscopic systems.
In the canonical ensemble, a system is in thermal equilibrium with a heat reservoir at a fixed temperature (T). The probability (P_i) of the system being in a microstate (i) with energy (E_i) is given by the Boltzmann distribution: [ P_i = \frac{e^{-E_i/k_B T}}{Z} ] where (Z = \sum_i e^{-E_i/k_B T}) is the canonical partition function and (k_B) is the Boltzmann constant. In this ensemble, thermodynamic quantities fluctuate because the system can ___ ________ ___ __________ ________ ______ _____.
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For the grand canonical ensemble, obtain an expression for the ensemble average energy and ensemble average particle numbers.
The time evolution of the quantum state, is given by the Schrödinger wave equation:
where H is the Hamiltonian operator. Using this equation, obtain time evolution of the wave function when the energy eigenvalue is E.
A proton inside a nucleus
may have speed upto
. How many quantum states are available to it?
State the Virial theorem and prove Boyle’s law using it.
What is meant by Cluster Integrals? Express B2 and B3 in terms of Cluster Integrals.
The time evolution of the quantum state, is given by the Schrödinger wave equation:
Using the relation of thermodynamic probability
Obtain an expression of Fermi Dirac distribution function. Show a plot of (the occupation index of a state corresponding to energy
) versus
for different temperatures and discuss it. What is the physical interpretation of Fermi energy?