A system of non-interacting Fermi particles with Fermi energy EF , has the density of states proportional to √E where E is the energy of a particles. The ratio of average energy per particle at T = 0 to the Fermi energy is.
We know that E = U/N
The correct answer is: 0.6
In a Maxwellian gas, if Vrms is the root means square velocity, then the most probable velocity vmp in terms of vrms is.
We know that
vmp = 0.8165 vrms
The correct answer is: 0.8165
A system has energy level E0, 2E0, 3E0,....., where the excited states are triply degenerated. Three non-interacting bosons are placed in the system. The total energy of these bosons is 5E0, the number of microstates is.
The total number of accessible states of non-interacting particles of spin 1/2 is given that N = 20.
Since the particles which exhibit odd integral spins are fermions and hence they obey Pauli’s exclusion principles. According to Pauli’s exclusion principle, not more than one particle can exist in the one quantum state. Hence, the number of assessible states N = 20.
The correct answer is: 20
The number of coordinates in the phase space of a single particle is.
For a single particle, these are 3 degrees of position coordinates x, y, z and 3 degrees of momentum coordinates px, py, pz, hence total coordinates = 6.
The correct answer is: 6
The total energy per particle of a collection of free fermions is 3eV. The Fermi energy of the system (in eV) is?
The correct answer is: 5
An ensemble of N three level systems with energies in thermal equilibrium at temperature T. Let If βε = 2, the probability of finding the system in level ε = 0 is given as (1 + 2cosh 2)α . Find value of α?
The probability of finding the system in the energy state with energy ε may be defined
But hence probability of finding the sytem with ε = 0 is
In question, βε0 = 2 so
P(0) = (1 + 2cosh 2)-1
⇒ α = 0.
The correct answer is: -1
The dimension of phase space of ten rigid diatomic molecule is
For a rigid diatomic molecule, there is no rotation possible along the inter-nuclear axis. That remove one angle from 3 Euler angles needed to describe the orientation of the molecules, and hence also removes an angular momentum component. That leaves 4 angular degrees of freedom plus 3 coordinate and 3 momenta for the center of mass motions.
∴ One diatomic molecule has phase space of dimension 10, so the number ofphase space for10 diatomic molecule = 10 ×10 = 100.
The correct answer is: 100
For a particle moving in a circle of fixed radius with constant speed, dimension of phase space is.
For a 2D moving in a plane as in case of a circular motion, we have two degrees of freedom for the q i.e. for x – y plane, q1 = x and q2 = y, and two degrees of freedom for the velocity given by ux and vy and thus px and py.
⇒ dimension of phase space = 2 + 2 = 4.
The correct answer is: 4
A system of N non-interacting classical point particles is constrained to move on the two-dimensional surface of a sphere. The internal energy of the system is in units of NkBT?
Since the particle is moving on two dimensional surface, its degree of freedom = 2.
Hence, for N non-interacting particles total degree of freedom = 2N
The energy per degree of freedom = 1/2 kBT
Hence, total energy
U = NkBT
The correct answer is: 1