In this article Operators of Quantum mechanics, We are going to discuss basic the information of operators, types of operators, properties of operators, and operation on operators.
Operators is a physical entity which operates on a function and converts it into another function.
Âψ = φ
Above expression shows that function φ is obtained as a result of applying an operation (an action) over the function ψ. This mathematical operation is denoted by Â. An operator is an instruction to carry out a certain mathematical operation on the symbol (function) following it. An operator is always placed on the left of the symbol (function) over which the operator has to give mathematical instruction.
Types of operators-
1) Linear operator – linear operator is an operator which satisfied the following condition-
Â(a ψ₁+ b ψ₂) = Â(aψ₁) + Â(b ψ₂)
Â(cψ₁) = c(Âψ₁)
2) Zero operator- Zero operators is an operator which apply on a function and convert it into zero function or vanish the function.
Ôψ = 0
3)Unit/identity operator- Unit or identity operator is an operator which operates on a function and makes no change. It is represented by Î
ÎÂ = ÂÎ = Â
4) Equal operators – Equal operator is an operator which satisfied the following condition-
Âψ = Êψ
Then Â and Ê are called equal operator.
5) unequal operator – Unequal operator is an operator which satisfied the following condition-
Âψ ≠ Êψ
Then Â and Ê are called unequal operators.
Properties of Operator
An operator is said to be a linear operator if it satisfied the following condition-
Â( ψ₁ + ψ₂ ) = Â ψ₁ + Â ψ₂
Âcψ = cÂψ
Where ψ₁ and ψ₂ are two functions and c is constant. there are two linear operators of fundamental importance.
(1) The null operator- if Âψ = 0 then Â = 0 , null operator.
(2)The unit operator – A operator which makes no change in function as discussed earlier.
Algebra of linear operators
(1) Sum of operators-
Ê = Â + Û is linear if,
Ê ψ= Â ψ+ Ûψ
Where ψ Is function.
Also, ( Â + Û) = ( Û+ Â ) which means that the sum of two operators is commutative.
Â = x and Û = d/dx
(x+ d/dx) ψ = xψ + dψ/dx (1)
(d/dx + x)ψ = dψ/dx + xψ (2)
both (1) and (2) shows identical results, which means the operators x and d/dx are commutative.
If ( Â + Û) + Ê = Â +( Û+ Ê)
Then the operator is said to be associative.
(2) If Â ( Ûψ)=Ê ψ, then Ê is the product of Â and Û. However, it should be remembered that it is not essential that the product ÂÛ should be equal to the product ÛÂ of the operator. In other words, the operators Â and Û may not commute. the product may be non-commutative
i.e. ÂÛ ≠ ÛÂ
For example, let Â= x and Û = d/dx
Then ( Â .Û)ψ = (xd/dx )ψ = xdψ/dx
And , ( Û .Ê) ψ = (dx/dx)ψ = d/dx (xψ) = ψ + x dψ/dx
( Û .Ê) ψ = ( 1 + xd/dx) ψ = (î + Â .Û)ψ
Which is true for every function of ψ. Hence in the present example, we find that the operator x and d/dx do not commute.
It should be remembered and well understood that for using the operator each operation is to be performed on the entire quantity placed on the right of the operator symbol.
The square of an operator Â is
Â² = Â.Â
Consider The following equation as an example for the square of an operator.
d²y/dx² + k²y =0
It is same as
(Â² + k²î) y =0
With Â= d/dx such that Â² = Â.Â
And ( Â.Â) y = (d/dx. d/dx)y = d/dx.dy/dx = d²y/dx²
Thus an algebraicbraic function of a linear operator is itself a linear operator.
The operator ( Â .Û- ÛÂ ) is called the commutator of the operator Â and Û and for brevity, it is written as [Â, Û ]
i.e. [Â ,Û ] = Â .Û- ÛÂ
If the commutator
[Â, Û ] =0, then the operators’ commute
Note- For a commutator of operators a square bracket [ ] is to be used.
Operator for energy and momentum
The wave function for a free particle in one dimension as obtained for Schrodinger’s equation was given by
ψ( x,t) = exp i/
Now, δψ/δt = iEψ/
h h δψ/δt = Eψ (1)
δψ/δx = i /
hδψ/δx = pψ (2)
From equation (1) and (2)for E and P, we infer that E and P can be represented by operators.
Operator for energy Ê = i
Operators of Quantum mechanics gives many quantitative information such as basic information about operators, their properties, and many other important data. Some of the important questions are given below-