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 _______ ___ ______ _______ ____ _____ _________ ________ _______ _________ _______ ________.
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Show under what condition Poisson distribution tends to Normal distribution.
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.
A monthly demand for commodity is a continuous random distribution with probability distribution function given as:
where N is normalization constant. Obtain the value of N so that the function is normalized and hence, find the mean and the variance.
Show that the probability that a state with energy above the Fermi level
filled is equal to the probability of a state with energy
below the Fermi level
is empty.
btain the phase space area enclosed by a classical harmonic oscillator for energies ranging from 0 to E.
State the Virial theorem and prove Boyle’s law using it.