11 Nature of Particles and Statistics:

Consider a gas of N non-interacting particles described by the Hamiltonian

where are the coordinate and momentum of the i th particle, is the Hamiltonian operator. A stationary system of N particles in a volume V then can be in any one of the quantum states determined by the solutions of the time independent Schödinger equation

where E is the eigenvalue of the Hamiltonian and is the corresponding eigenfunction.

If there are particles in an eigenstate , then the distribution should satisfy

and the wave functions for N particles with particle in the th state can be written as

(i)

(ii)

where is the eigenfunction of the single particle Hamiltonian with eigenvalue . Each single particle wave function is always a linear combination of a set of orthonormal basis functions , . The particles described by these three wave functions obey different statistics. (i)The particles described by the product function correspond to different microstate by interchanging particles between states. These are then distinguishable particles and obey Maxwell-Boltzmann statistics. In the case of symmetric wave functions, interchanging of particles does not generate a new microstate. Thus, the particles are indistinguishable. Also, all the particles in a single state correspond to a non-vanishing wave function. That means accumulation of all the particles in a single state is possible. These particles obey Bose-Einstein statistics and are called bosons . For the anti-symmetric wave function, if the two particles are exchanged, the two columns of the determinant are exchanged and leads to the same wave function with a different sign. Thus, the particles are again indistinguishable. However, if any two particles are in one state then the corresponding rows of the determinant are the same and the wave function vanishes. This means that a state cannot be occupied by more than one particle. This is known as Pauli principle. These particles obey Fermi-Dirac statistics and they are called fermions .