Intrinsic parity

In quantum mechanics, the intrinsic parity is a phase factor that arises as an eigenvalue of the parity operation x i → x i ′ = − x i {displaystyle x_{i} ightarrow x_{i}'=-x_{i}} (a reflection about the origin). To see that the parity's eigenvalues are phase factors, we assume an eigenstate of the parity operation (this is realized because the intrinsic parity is a property of a particle species) and use the fact that two parity transformations leave the particle in the same state, thus the new wave function can differ by only a phase factor, i.e.: P 2 ψ = e i ϕ ψ {displaystyle P^{2}psi =e^{iphi }psi } thus P ψ = ± e i ϕ / 2 ψ {displaystyle Ppsi =pm e^{iphi /2}psi } , since these are the only eigenstates satisfying the above equation. In quantum mechanics, the intrinsic parity is a phase factor that arises as an eigenvalue of the parity operation x i → x i ′ = − x i {displaystyle x_{i} ightarrow x_{i}'=-x_{i}} (a reflection about the origin). To see that the parity's eigenvalues are phase factors, we assume an eigenstate of the parity operation (this is realized because the intrinsic parity is a property of a particle species) and use the fact that two parity transformations leave the particle in the same state, thus the new wave function can differ by only a phase factor, i.e.: P 2 ψ = e i ϕ ψ {displaystyle P^{2}psi =e^{iphi }psi } thus P ψ = ± e i ϕ / 2 ψ {displaystyle Ppsi =pm e^{iphi /2}psi } , since these are the only eigenstates satisfying the above equation. The intrinsic parity's phase is conserved for non-weak interactions (the product of the intrinsic parities is the same before and after the reaction). As [ P , H ] = 0 {displaystyle =0} the Hamiltonian is invariant under a parity transformation. The intrinsic parity of a system is the product of the intrinsic parities of the particles, for instance for noninteracting particles we have P ( | 1 ⟩ | 2 ⟩ ) = ( P | 1 ⟩ ) ( P | 2 ⟩ ) {displaystyle P(|1 angle |2 angle )=(P|1 angle )(P|2 angle )} . Since the parity commutes with the Hamiltonian and d P d t = 0 {displaystyle {frac {dP}{dt}}=0} its eigenvalue does not change with time, therefore the intrinsic parities phase is a conserved quantity. A consequence of the Dirac equation is that the intrinsic parity of fermions and antifermions obey the relation P f ¯ P f = − 1 {displaystyle P_{ar {f}}P_{f}=-1} , so particles and their antiparticles have the opposite parity. Single leptons can never be created or destroyed in experiments, as lepton number is a conserved quantity. This means experiments are unable to distinguish the sign of a leptons parity, so by convention it is chosen that leptons have intrinsic parity +1, antileptons have P = − 1 {displaystyle P=-1} . Similarly the parity of the quarks is chosen to be +1, and antiquarks is -1.