| Parity (physics) |
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: A 3×3 matrix representation of P would have Determinant equal to –1, and hence cannot reduce to a Rotation . In a two-dimensional plane, parity is the same as a Rotation by 180 degrees. SIMPLE SYMMETRY RELATIONS Under Rotation s, classical geometrical objects can be classified into Scalar s, Vector s, and Tensor s of higher rank. In Classical Physics , physical configurations need to transform under Representation s of every symmetry group. In a Quantum Theory states in a Hilbert Space do not need to transform under Representation s of the Group of rotations, but only under Projective Representation s. 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 an 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. Projective representations of the rotation group that are not representations are called Spinor s, 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
One can define reflections such as : which also have negative determinant. Then, combining them with rotations one can generate the parity transformation defined earlier. In any even number of dimensions, the first definition of parity has positive determinant, and hence can be obtained as some rotation. One then uses reflections to extend the notion of scalars and vectors to pseudo-scalars and pseudo-vectors. Parity forms the . 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 . CLASSICAL MECHANICS Newton's equation of motion F = ma (if mass is constant) 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. :L = '''r''' × '''p''' , :P('''L''') = (–'''r''') × (–'''p''') = '''L'''. In classical Electrodynamics , charge density ρ is a scalar, the electric field, '''E''', and current '''j''' are vectors, but the magnetic field, '''H''' is an axial vector. However, Maxwell's equations are invariant under parity because the curl of an axial vector is a vector. QUANTUM MECHANICS Possible eigenvalues are given by a pair of quantum states which go into each other under parity. However, this representation can always be reduced to linear combinations of states, each of which is either even or odd under parity. One says that all Irreducible Representation s of parity are one-dimensional.]] In Quantum Mechanics , spacetime transformations act on Quantum States . The parity transformation, P, is a Unitary Operator in quantum mechanics, acting on a state '''ψ''' as follows: P '''ψ'''(r) = '''ψ'''(-r). One must have P2 '''ψ'''(r) = ei φ '''ψ'''(r), since an overall phase is unobservable. The operator P2, which reverses the parity of a state twice, leaves the spacetime invariant and so is an internal symmetry which rotates its eigenstates by phases ei φ. If P2 is an element ei Q of a continuous U(1) symmetry group of phase rotations then e-i Q/2 is part of this U(1) and so is also a symmetry. In particular we can define P'=Pe-i Q/2 which is also a symmetry and so we can choose to call P' our parity operator instead of P. Notice that P'2=1 and so P' has eigenvalues ±1. However when no such symmetry group exists, it may be that all parity transformations have some eigenvalues which are phases other than ±1. 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, Hamiltonian s are Invariant (symmetric) under a parity transformation if P commutes with the Hamiltonian. In non-relativistic quantum mechanics, this happens for any potential which is scalar, ie, '''V''' = '''V'''(r), hence the potential is spherically symmetric. The following facts can be easily proven: | ||
| For A State ''''''L,m'''>''' Of Orbital Angular Momentum '''L''' With Z-axis Projection '''m''', '''P''' ''''''L,m'''>'''  |  (-1)<sup>L<sup>''''''L, m'''>''' |
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| If We Can Show That The | "http://wwwinformationdelightinfo/encyclopedia/entry/vacuum_state" class="copylinks">Vacuum State is invariant under parity ('''P''' ''''''0'''>''' = ''''''0'''>'''), the Hamiltonian is parity invariant (''' {Link without Title} ''' = '''0''') and the quantization conditions remain unchanged under parity, then it follows that every state has good parity, and this parity is conserved in any reaction |
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