Pauli's Exclusion Principle & Symmetric and Antisymmetric Wavefunction.

Pauli's Exclusion Principle;

Pauli found that in a quantum mechanical system, no two electrons can occupy the same state, i.e no two electrons can have the same set of quantum number n,l,mₗ and mₛ. With the help of this principle, one can explain the construction of the periodic table.

Let us consider a system of two non-interacting identical particles. Let the particle 1 be in a quantum state denoted by a, so that its wavefunction is

Also, the particle 2 is in quantum state b so that its wavefunction is

The total wavefunction is given by

When the particles are exchanged then the total wavefunction is given by

We have two types of the wavefunctions for a system of two particles.

Symmetric and Antisymmetric Wavefunction;

Now the wavefunction which do not change sign on exchange of particles are called symmetric wavefunction and those which change their sign on exchange of particles are called anti-symmetric wavefunction.

Symmetric wavefunction:

For a symmetric wavefunction

The linear combination of these two wavefunctions gives us the symmetric wavefunction

linear combination of the two wavefunctions. Therefore, for symmetric wavefunctions

Where the factor 1/√2 is required to normalize the wavefunction 𝜳ₛ.

Anti-symmetric wavefunction:

For an anti-symmetric wavefunction

The linear combination of these two wavefunctions gives the anti-symmetric wavefunction.

If a=b i.e., the two identical particles have same quantum state, then the resultant state disappears.

i.e 𝜳A =0

Hence we conclude that the probability of two anti-symmetric identical particles is zero or two non-interacting fermions cannot be both in the same quantum state. This is Pauli exclusion principle.

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