A Identical Fermions in identical linear combinations

[hokhani]

According to Pauli principle the two fermions can not occupy one state of a Hamiltonian. Can the two fermions occupy a state which is linear recombination of two states of the Hamiltonian?

 

[DrClaude]

According to Pauli principle the two fermions can not occupy one state of a Hamiltonian. Can the two fermions occupy a state which is linear recombination of two states of the Hamiltonian?

Actually, the Pauli principle says that the wave function of two identical fermions must be anti-symmetric with respect to the interchange of the two fermions. This leads to the Pauli exclusion principle, as it is impossible to built an anti-symmetric state if the two fermions are in the same state. Hence, it can be a superposition of eigenstates, so long as that superposition is anti-symmetric.

 

[hokhani]

Hence, it can be a superposition of eigenstates, so long as that superposition is anti-symmetric.

Thanks. If we consider the superposition as \phi, according to your statement the two fermions (disregarding spin) can never settle in \phi because \phi(1) \phi(2)=\phi(2) \phi(1). Is it right?

 

[DrClaude]

Thanks. If we consider the superposition as \phi, according to your statement the two fermions (disregarding spin) can never settle in \phi because \phi(1) \phi(2)=\phi(2) \phi(1). Is it right?

Yes, that state is symmetric, and thus is a valid state for two identical bosons only.