Parity
One of the fundamental nuclear properties is parity. According to quantum mechanics, a wave is associated with a moving particle. This wave corresponds to a wave function \( \psi \) which depends on co-ordinates \((x, y, z)\). The wave function \( \psi \) may be regarded as a measure of the probability of a particle around a particular position. The probability of finding a particle is given by:
\[ |\psi|^2 = \psi \psi^* \]
where \( \psi^* \) is the complex conjugate of \( \psi \).
If the particle is definitely to be found in a distance \( dx \), the probability of finding the particle in distance \( dx \) must be unity.
In general
\[ \int |\psi|^2 \, dx = 1 \]
Parity is considered to be more important than nuclear spin. Parity of a nucleus refers to the behaviour of this wave function \( \psi \) under inversion of co-ordinates i.e. If we replace \( x \) by \( -x \), \( y \) by \( -y \) and \( z \) by \( -z \), then the wave function is said to possess an even parity if
\[ \psi(x, y, z) = \psi(-x, -y, -z) \]
and odd parity if
\[ \psi(x, y, z) = -\psi(-x, -y, -z) \]
In general
\[ \psi(x, y, z) = P \psi(-x, -y, -z) \]
where \( P = \pm 1 \).
\( P \) can be taken as a quantum number and the property defined by it is called parity of the system.
\( P = +1 \) gives even parity.
\( P = -1 \) gives odd parity.
Let us take the example of a system with potential energy as a symmetric function of co-ordinates.
In the case symmetry of wave function is
\[ V(x, y, z) = V(-x, -y, -z) \]
Nuclear states are characterized by a definite parity which may be different for different states of the same nucleus. Parity remains conserved during the nuclear transformations.
Parity of a nucleus is closely associated with the value of orbital angular momentum \( L \).
For \( L = 0, 2, 4, \ldots \), parity is even and if \( L \) is odd, parity is odd. It is given by
\[ P = (-1)^L \]
Thus if \( \psi \) is the wave function of a state with parity sign and parity is positive or negative, then the wave function \( \psi \) may or may not change sign on inversion of co-ordinates. It may be pointed out that parity is a multiplicative quantum number.
Thus if two systems with wave functions \( \psi_1 \) and \( \psi_2 \) are combined, the parity \( P \) of the system is the product of the parities \( P_1 \) and \( P_2 \) of the two systems.
Parity is a purely quantum mechanical concept having no classical analogue.