State the Virial theorem and prove Boyle’s law using it.
The Virial theorem states that for a stable system of particles bound by potential forces, the time average of the total kinetic energy ( \langle T \rangle ) is related to the time average of the total potential energy ( \langle V \rangle ) as: [ 2 \langle T \rangle = \sum_i \langle \vec{r}_i \cdot \vec{F}_i \rangle ] For a ________ _____ _____ _______ ______ _______ ____ _____ ____ __________ _________.
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What is the Bose-Einstein condensation? Write an expression of number of particles in the excited state (Nex) in terms of Bose-Einstein condensation temperature (Tc), total number of bosons (N) in an assembly and temperature T. Show a plot of distribution of bosons as a function of temperature below and above the Tc.
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.
For the grand canonical ensemble, obtain an expression for the ensemble average energy and ensemble average particle numbers.
Using entropy as a function of temperature and pressure, obtain the first Ehrenfest’s equation.
State and prove quantum Liouville equation.
The time evolution of the quantum state, is given by the Schrödinger wave equation:
N particles obey Maxwell-Boltzmann distribution. They are distributed among three states with energies and
. If the equilibrium energy of the system is approximately 3000kBT, calculate the total number of particles.