Parity (physics)


In physics, a parity transformation is the flip in the sign of one spatial coordinate. In three dimensions, it can also refer to the simultaneous flip in the sign of all three spatial coordinates :
It can also be thought of as a test for chirality of a physical phenomenon, in that a parity inversion transforms a phenomenon into its mirror image.
All fundamental interactions of elementary particles, with the exception of the weak interaction, are symmetric under parity transformation. As established by the Wu experiment conducted at the US National Bureau of Standards by Chinese-American scientist Chien-Shiung Wu, the weak interaction is chiral and thus provides a means for probing chirality in physics. In her experiment, Wu took advantage of the controlling role of weak interactions in radioactive decay of atomic isotopes to establish the chirality of the weak force.
By contrast, in interactions that are symmetric under parity, such as electromagnetism in atomic and molecular physics, parity serves as a powerful controlling principle underlying quantum transitions.
A matrix representation of P has determinant equal to −1, and hence is distinct from a rotation, which has a determinant equal to 1. In a two-dimensional plane, a simultaneous flip of all coordinates in sign is not a parity transformation; it is the same as a 180° rotation.
In quantum mechanics, wave functions that are unchanged by a parity transformation are described as even functions, while those that change sign under a parity transformation are odd functions.

Simple symmetry relations

Under rotations, classical geometrical objects can be classified into scalars, vectors, and tensors of higher rank. In classical physics, physical configurations need to transform under representations of every symmetry group.
Quantum theory predicts that states in a Hilbert space do not need to transform under representations of the group of rotations, but only under projective representations. The word projective refers to the fact that if one projects out the phase of each state, where we recall that the overall phase of a quantum state is not observable, then a projective representation reduces to an ordinary representation. All representations are also projective representations, but the converse is not true, therefore the projective representation condition on quantum states is weaker than the representation condition on classical states.
The projective representations of any group are isomorphic to the ordinary representations of a central extension of the group. For example, projective representations of the 3-dimensional rotation group, which is the special orthogonal group SO, are ordinary representations of the special unitary group SU. Projective representations of the rotation group that are not representations are called spinors and so quantum states may transform not only as tensors but also as spinors.
If one adds to this a classification by parity, these can be extended, for example, into notions of
  • scalars and pseudoscalars which are rotationally invariant.
  • vectors and axial vectors which both transform as vectors under rotation.
One can define reflections such as
which also have negative determinant and form a valid parity transformation. Then, combining them with rotations one can recover the particular parity transformation defined earlier. The first parity transformation given does not work in an even number of dimensions, though, because it results in a positive determinant. In even dimensions only the latter example of a parity transformation can be used.
Parity forms the abelian group due to the relation. All Abelian groups have only one-dimensional irreducible representations. For, there are two irreducible representations: one is even under parity,, the other is odd,. These are useful in quantum mechanics. However, as is elaborated below, in quantum mechanics states need not transform under actual representations of parity but only under projective representations and so in principle a parity transformation may rotate a state by any phase.

Representations of O(3)

An alternative way to write the above classification of scalars, pseudoscalars, vectors and pseudovectors is in terms of the representation space that each object transforms in. This can be given in terms of the group homomorphism which defines the representation. For a matrix
  • scalars:, the trivial representation
  • pseudoscalars:
  • vectors:, the fundamental representation
  • pseudovectors:
When the representation is restricted to, scalars and pseudoscalars transform identically, as do vectors and pseudovectors.

Classical mechanics

Newton's equation of motion equates two vectors, and hence is invariant under parity. The law of gravity also involves only vectors and is also, therefore, invariant under parity.
However, angular momentum is an axial vector,
In classical electrodynamics, the charge density is a scalar, the electric field,, and current are vectors, but the magnetic field, is an axial vector. However, Maxwell's equations are invariant under parity because the curl of an axial vector is a vector.

Effect of spatial inversion on some variables of classical physics

The two major divisions of classical physical variables have either even or odd parity. The way into which particular variables and vectors sort out into either category depends on whether the number of dimensions of space is either an odd or even number. The categories of odd or even given below for the parity transformation is a different, but intimately related issue.
The answers given below are correct for 3 spatial dimensions. In a 2 dimensional space, for example, when constrained to remain on the surface of a planet, some of the variables switch sides.

Odd

Classical variables whose signs flip under space inversion are predominantly vectors. They include:

Even

Classical variables, predominantly scalar quantities, which do not change upon spatial inversion include:

Quantum mechanics

Possible eigenvalues

In quantum mechanics, spacetime transformations act on quantum states. The parity transformation,, is a unitary operator, in general acting on a state as follows:.
One must then have, since an overall phase is unobservable. The operator, which reverses the parity of a state twice, leaves the spacetime invariant, and so is an internal symmetry which rotates its eigenstates by phases. If is an element of a continuous U symmetry group of phase rotations, then is part of this U and so is also a symmetry. In particular, we can define, which is also a symmetry, and so we can choose to call our parity operator, instead of. Note that and so has eigenvalues. Wave functions with eigenvalue under a parity transformation are even functions, while eigenvalue corresponds to odd functions. However, when no such symmetry group exists, it may be that all parity transformations have some eigenvalues which are phases other than.
For electronic wavefunctions, even states are usually indicated by a subscript g for gerade and odd states by a subscript u for ungerade. For example, the lowest energy level of the hydrogen molecule ion is labelled and the next-closest energy level is labelled.
The wave functions of a particle moving into an external potential, which is centrosymmetric, either remain invariable or change signs: these two possible states are called the even state or odd state of the wave functions.
The law of conservation of parity of particles states that, if an isolated ensemble of particles has a definite parity, then the parity remains invariable in the process of ensemble evolution. However this is not true for the beta decay of nuclei, because the weak nuclear interaction violates parity.
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, one can always take linear combinations of quantum states such that they are either even or odd under parity. 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 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 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

s have parity 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. However the third excited term at about 83,300 cm−1 above the ground state has electron configuration 1s22s22p23s has even parity since there are only two 2p electrons, and its term symbol is 4P.