ensemble evolution. However this is not true for the beta decay of nuclei) which is due to the weak nuclear interaction. The parity of the states of a particle moving in a spherically symmetric external field is determined by the angular momentum, and the particle state is defined by three quantum numbers: total energy, angular momentum and the projection of angular momentum. Consequences of parity symmetry When parity generates the Abelian group ℤ2, one can always take linear combinations of quantum states such that they are either even or odd under parity (see the figure). Thus the parity of such states is ±1. The parity of a multiparticle state is the product of the parities of each state; in other words parity is a multiplicative quantum number. In quantum mechanics, Hamiltonians are invariant (symmetric) under a parity transformation if commutes with the Hamiltonian. In non-relativistic quantum mechanics, this happens for any scalar potential, i.e., , hence the potential is spherically symmetric. The following facts can be easily proven: If and have the same parity, then where is the position operator. For a state of orbital angular momentum with z-axis projection , then . If , then atomic dipole transitions only occur between states of opposite parity. If , then a non-degenerate eigenstate of is also an eigenstate of the parity operator; i.e., a non-degenerate eigenfunction of is either invariant to or is changed in sign by . Some of the non-degenerate eigenfunctions of are unaffected (invariant) by parity and the others are merely reversed in sign when the Hamiltonian operator and the parity operator commute: where is a constant, the eigenvalue of , Many-particle systems: atoms, molecules, nuclei The overall parity of a many-particle system is the product of the parities of the one-particle states. It is −1 if an odd number of particles are in odd-parity states, and +1 otherwise. Different notations are in use to denote the parity of nuclei, atoms, and molecules. Atoms Atomic orbitals have parity (−1)ℓ, where the exponent ℓ is the azimuthal quantum number. The parity is odd for orbitals p, f, ... with ℓ = 1, 3, ..., and an atomic state has odd parity if an odd number of electrons occupy these orbitals. For example, the ground state of the nitrogen atom has the electron configuration 1s22s22p3, and is identified by the term symbol 4So, where the superscript o denotes odd parity.