QUANTUM MECHANICS

Quantum Mechanics

[Institution name]

Quantum Mechanics

Maxwell-Boltzmann statistics are generally used in order to predict what proportion exists in any given energy level, given a large number of atoms within a thermal equilibrium. For example, proportion of energy in the ground state and the level of energy in the first excited state.

When applying the Maxwell-Boltzmann statistics on a container full of an ideal gas, the relevant energy levels are obtained with the practice of solving time-dependent Schrödinger equation. This is necessary to assess the atom confined inside that particular container. Therefore, the given probability that an atom will be occupying in the nth state is proportional to exp [-En/(kßT)] Maxwell-Boltzmann statistics predict that the number of particles with energy, tj, is Nj:

Where Ni= number of particles in state i

?i = energy of the ith state

gi = the degeneracy of state I, number of microstates with energy ?i

µ = the chemical potential

k = Boltzmann's constant

T = the absolute temperature

N = the total number of particles

Z = the partition function

The statistics of Maxwell-Boltzmann are also referred as distinguishable classical particles statistics. For instance, you have a collection of 10 particles with only 2 possible energy states available to be occupied. This is restricted to the condition that the particles are distinguishable. This condition applies that you can put labels on them (particle #1, particle #2, etc). if particles #1,#2,#3,#4,#5 are in state 1, and #6,#7,#8,#9,#10 are in state 2, that is a different distribution than if you were to switch, say, #6 and #1. There are 210 = 1024 ways of arranging these 10 particles among 2 states. The statistics of Maxwell-Boltzmann are specifically useful in situations when dealing with gases that have low density and relatively high temperatures to meet the specific criteria.

Fermi Dirac Statistics: There is one of the two possible ways to refer a system that deals with indistinguishable particles that may be distributed among a set of energy states. Availability of each of the state can be occupied by only one particle. This characteristic accounts for atom's electron structure which remains in separate states rather than colliding into a single state within some aspects of electrical conductivity.

The Fermi-Dirac statistics is applicable only to specific particles they obey the rules and limitations known as Pauli Exclusion Principle. Such kind of particles has semi-integer value of spin and termed as fermions, as this statistics accurately describes their behaviour. For example, Fermi-Dirac applies to electrons, neutrons, and protons. The expected number of particles, n, in states with energy ?i is mentioned as follows:

Where: n, = the number of particles in state i

?i = the energy of state i

gi = the degeneracy of state i, ( the number of microstates with energy ?i)

µ = the chemical potential (sometimes the Fermi energy is used instead)

k = Boltzmann's constant

T = the absolute temperature

The statistics applies to the fermions and most often used for studying electrons present in solids. For this reason, they are considered as the basis for electronics and ...