The Hamiltonian for the non-relativistic, non-interacting, monoatomic ideal gas is given by
where m is the mass of the particle and U(qi) is the potential energy. Obtain an expression of the phase space volume of the energy shell and hence obtain an expression of number of microstates.
The Hamiltonian of a non-relativistic, non-interacting, monoatomic ideal gas is [ H(q,p) = \sum_{i=1}^{3N} \frac{p_i^2}{2m} + U(q_i), ] where (p_i) are the momenta, (q_i) are the positions, and (U(q_i)) is the potential energy. The phase space volume corresponding to energies less than or equal to (E) is defined as [ \Omega(E) = \frac{1}{h^{3N}} \int \cdots \int \mathrm{d}^{3N}q , \mathrm{d}^{3N}p ; \Theta(E - H(q,p)), ] where ______ __________ _______ _______ ________.
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Show under what condition Poisson distribution tends to Normal distribution.
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.
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.
State the Virial theorem and prove Boyle’s law using it.
A proton inside a nucleus
may have speed upto
. How many quantum states are available to it?
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.