Parity is also known as Multiplicative Quantum Number. Parity is a useful concept in both Nuclear Physics and Quantum Mechanics.
Parity helps us explain the type of stationary wave function (either symmetric or asymmetric) that subatomic particles, like neutrons, electrons, or protons have.
In simple words, parity is the reflection of coordinates about the origin. For instance, the wave functions of x, y, and z are ψx, y, z.
Here, the wave function, ψ explains the stationary state of any particle, and x, y, z are the functions of the position.
Further, the parity is the transpose of x, y, z as;
x → - x…..(1)
y → - y ….(2)
z → - z….(3)
To understand parity in particle physics in detail, view this page till the end.
Conservation of Parity in Particle Physics
Now, let’s understand the parity conservation in particle physics from the above three equations:
Assume that there is a three-axis coordinate system with x, y, and as coordinates.
A point P lying on the coordinate plane has a position of P (x, y, z). Suppose that we shift the position of P to another point that is at its mirror image.
Now, the coordinates of this point become P (- x, - y, - z), i.e., the transpose of the original coordinates. We call this practice the transformation or reflection about the origin.
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So, the changes we have made above are called Parity.
Certainly, we can state the definition of Parity into words:
Above all, Parity is the reflection of the coordinates in a plane around the origin. Furthermore, parity helps us define the stationary state of the wave function.
Further, parity can be explained in simple terms as;
Parity Particle in Physics
Now, let’s say, a parity operator “UP,” where “U” is the operator and “P” is parity. Further, this parity operator must follow the below property:
Similarly, the unity operator property is:
UP* UPt = 1….(b)
Indeed, the above equation (b) says that the product of the operator and the transpose of an operator is always unity.
Now, comparing equation (1) and (2), we get:
As a matter of fact, the transpose of an operator becomes equal to the operator itself, we call this property the Hermitian Property or Hermitian Operator.
Points To Note:
So, when the above two conditions are equal, we get what we call the “Hermitian Operator.”
Besides, the condition of the Hermitian operator says that all its eigenvalues are real.
Now, let’s understand the Parity of Elementary Particle:
Parity of Elementary Particle
Let’s say, a wave function ψx, y, z = ψ(R), where the function “R = x, y, z.”
Furthermore, when the wave function is operated in the following manner, we get an eigenvalue “P;”
UPψ(R) = P ψ(R)......(c)
Additionally, operating the function, we may or may not get the same function. However, in the above case, the function is the same, i.e., ψ(R).
Now, again operating the above equation (c) with a parity operator as;
UP(UPψ(R)) = UP(P ψ(R)) …..(d)
We know that the eigenvalue is always real and constant, so taking “P” out from equation (d), we have:
UP(UPψ(R)) = P . UP(ψ(R))
As we know from equation (c), UPψ(R) = P ψ(R), putting the same in equation (e), we get:
UP(UPψ(R)) = P . P ψ(R)
UP2 ψ(R) = P2 ψ(R)
Since P is an eigenvalue and P2 = 1,therefore,
The above equation (f) has two meanings, let’s understand these:
P = + 1
P = - 1
What is the Significance of a Parity Operator?
Now, let’s see what even and odd parity is.
Considering a function, ψx, y, z on transposing, forms ψ- x, - y, - z.
On doing the transpose of ψ- x, - y, - z, we get “ψx, y, z“ again, which means, it is an even function.
However, if we get (- ψx, y, z ) in place of (ψx, y, z), it is an odd function.
When the sign of function remains the same, it is an even parity, i.e., (+ ψx, y, z).
Furthermore, when the sign of function remains the same, it is an odd parity,
i.e., (- ψx, y, z ).
Parity Signs Explained
Let’s suppose that we change the coordinates of the stationary state of the particle.
However, on changing if we get the same function, it is an even parity, such as (ψ- x, - y, - z) remains (+ ψx, y, z) after a change. We call this function the symmetric function.
Moreover, if we get a different function, i.e., from (ψ- x, - y, - z) to ((+ ψx, y, z) after a change. We call this function an asymmetric function.