Parity

In Physics, parity is a feature that is significant in describing a physical system using quantum mechanics. It usually has to do with the symmetry of the wave function that represents a system of fundamental particles. A parity transformation is a type of mirror image that substitutes such a system. In mathematical terms, the system's spatial coordinates are inverted through the origin point; that is, the coordinates x, y, and z are substituted with x, y, and z. In general, a system is said to have even parity if it is identical to the original system following a parity transformation. Its parity is odd if the final formulation is the inverse of the original. Physical observables that are dependent on the square of the wave function are unaffected by parity. The overall parity of a complex system is the product of the parties of its constituents.

Conservation of Parity

A Physics principle says that two mirror images of each other, such as left-spinning and right-spinning particles, should act the same. The idea does not apply to subatomic particle interactions that are weak.

Fundamental Particles Sign

In the study of basic particles, there have been some remarkable breakthroughs in recent years. One of the outcomes has been the appearance of a new language and a huge number of new symbols in scientific writing. Specific types of particles are designated by symbols such as,, and. Others have been used to characterize only phenomenological behaviour (e.g.,). Various authors have given different names to the same particle or assigned different meanings to the same symbol. The meaning of a sign can sometimes alter throughout time. To offer an example, the Greek letter was originally employed to denote a heavy meson that stopped in the emulsion and then decayed, resulting in a single ionising particle. Later, the Latin letter K was substituted for the Greek letter as a code for the above-mentioned phenomenological description.

While the letter took on a more concrete physical meaning: a hefty meson that decays into one charged and two neutral particles. The letter K is also often used to represent any charged particle that is heavier than a -meson but lighter than a proton and whose method of disintegration is unclear. Another example is the neutral particle with a mass of roughly 1,000 m e that decays into two -mesons, which has been given numerous names such as v0, V 20, and V 40, although some authors have used the letter V 20 to identify any V0-particle other than the so-called V 10. 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:

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 equations (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:

• 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(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)

So, 

                  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:

• 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.