# Parity

## What is Parity?

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.

So, the changes we have made above are called Parity.

Certainly, we can state the definition of Parity into words:

### Parity Definition

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:

UP2  ….(a)

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:

 UPt   =   UP

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:

• The value of parity operator UP must be one.

• Also, the parity operator must be a unitary operator.

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(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)

So,

UP2 ψ(R)P2 ψ(R)

Since P is an eigenvalue and P2 = 1,therefore,

 P    = $\pm1$ ……(f)

The above equation (f) has two meanings, let’s understand these:

• Even Parity

P = + 1

• Odd Parity

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.

Q1: Define the Term Parity.

Ans: In physics, parity is an important concept in the quantum-mechanical description of a physical system.

Moreover, it relates to the symmetry of the wave function representing a fundamental particle system. A parity transformation exchanges such a system with a reflection/mirror image.

Q3: What is the Best Description of Quantum Mechanics.

Ans: Quantum mechanics deals with the behaviour of matter and light on the atomic and subatomic scale.

Above all, it attempts to describe and account for the properties of molecules and atoms and their constituents, viz: electrons, protons, neutrons, and additional esoteric particles like quarks and gluons.

Hence, quantum mechanics help mathematical machines presume the behaviours of microscopic particles, furthermore, of the measuring instruments we use to investigate these behaviours - and in this capacity, it is remarkably successful: in terms of power and precision.

Q3: Are Hermitian Operators Real?

Ans: Hermitian operators have the following attributes:

• Real eigenvalues

• Orthogonal eigenfunctions, and

• Corresponding eigenfunctions

The three forms a complete biorthogonal system that is second-order and linear.

Additionally, the concept of the Hermitian operator has an extension in quantum mechanics to operators that need be neither second-order differential nor real.

Q4: What is Parity in Classical Mechanics?

Ans: In classical mechanics, the parity changes the position vector from x → − x.

Furthermore, operators (such as O) after changing under symmetry operators are represented by

Q as O → Q − 1ˆO Q.

Comment