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2.6 Identical Particles and Exchange Degeneracy

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2.6 Identical Particles and Exchange Degeneracy 2.6 Identical particles and exchange degeneracy • Consider a system of n particles each having its own coordinates. • If interchange of any two or more particles leaves the system unchanged, the particles are said to be identical or indistinguishable particles. • In other words, the Hamiltonian of the system remains symmetric with respect to the interchange of the coordinates of any two particles. Mathematically it is written as H ,3,2,1( LL n), = H ,3,1,2( LL n), • For simplicity we consider a system of two particles. We can define an exchange operator P12 which when operates on a wave function, interchanges the coordinates of two particles i.e. ψ =ψ P12 )2,1( )1,2( Operating P12 twice on the wave function brings it back to its original value. Thus the wave function can change at most by a sign. ψ ±= ψ P12 )2,1( )1,2( (2.132) • If the wave function does not change its sign under the exchange of two particles, it is said to be symmetric wave function ψsym . • On the other hand, if the wave function changes its sign under the exchange of two particles, it is said to be antisymmetric wave function ψas . • The Schrödinger equation for two particle system is ∂ H )2,1( Ψ t),2,1( = ih Ψ t),2,1( ∂t and iEt Ψ t),2,1( =ψ )2,1( exp − h 1 where E is the energy eigenvalue. • After the exchange of two particles, the resulting wavefunction ψ(2, 1) is also an eigenfunction of the Hamiltonian with energy eigenvalue E. • Similarly, for a system with n particles, n! Solutions are possible with same energy eigenvalue E. • This degeneracy arising due to interchange of identical particles is called the exchange degeneracy . • Since ψ(1, 2) and ψ(2, 1) both are solutions of the Schrödinger equation, their linear combination is also a solution. • Since particles are identical and we do not know which particle is staying in which state, therefore it is justified also to consider all possible combinations of particle exchange in the wave function. • In the light of above discussion, the symmetric and antisymmetric wave function are given as ψ = ψ )2,1( +ψ )1,2( (Symmetric wavefunct ion) sym (2.133) ψ = ψ −ψ as )2,1( )1,2( (Antisymme tric wavefunct ion) 2 2.6.1 Wave function for many electron system • Considering noninteracting identical particles say electrons. The Hamiltonian is given as H )2,1( = H )1( + H )2( • The energy eigenfunction is given as product of single particle eigenfunctions as ψ = φ φ )2,1( a )1( b )2( where electron ‘1’ occupies state φa and electron ‘2’ occupies state φb. The single particle eigenfunctions follow the relation H )1( φ )1( = E φ )1( a aa (2.134) φ = φ H )2( b )2( E bb )2( where Ea and Eb are the energy eigenvalues corresponding to single particle states. • The total energy eigenvalue of the system is = + E Ea Eb • Since the electrons are indistinguishable, we can only say that one electron is occupying one state but we can not say which electron is staying in which state. • φ φ Thus for a system of two electrons, the possible eigenfunctions are a )1( b )2( and φ φ a )2( b )1( . The symmetric and antisymmetric combinations are ψ = φ φ +φ φ sym a )1( b )2( a )2( b )1( (2.135) φ )1( φ )2( ψ = φ )1( φ )2( − φ )2( φ )1( ≡ a a (2.136) as a b a b φ φ b )1( b )2( • If both the electrons are occupying same state then according to eqs. (2.135) and (2.136), we have ψ = φ φ + φ φ = φ φ sym a )1( a )2( a )2( a )1( 2 a )1( a )2( (2.137) ψ = φ φ − φ φ = as a )1( a )2( a )2( a )1( 0 (2.138) 3 • The symmetry and antisymmetry is an important property of the wave function and is related to the spin of the identical particles. Pauli demonstrated that: (i) A system containing identical particles with half integer spin (2 n+1)/2, where n = ,2,1,0 L, are described by antisymmetric wave functions. Such particles are called ‘ fermions ’ and obey Fermi-Dirac statistics. (ii) A system containing identical particles with integer spin ( ,2,1,0L ) are described by symmetric wave functions. Such particles are called ‘ bosons ’ and obey Bose-Einstein statistics. • The eq. (2.138) is in accordance with the Pauli’s exclusion principle which states that two identical fermions can not occupy the same state. • Generalizing to n electron system, the normalized antisymmetric wave function of the system is given as φ φ L φ a )1( a )2( a n)( 1 φ )1( φ )2( L φ n)( ψ = b b b (2.139) as n M M M φ φ L φ n )1( n )2( n n)( where /1n is the normalization factor and the determinant is called the Slater determinant . 4 .
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