Operators in Quantum Mechanics

Operators in Quantum Mechanics

An operator is a symbol for an instruction to carry out some action, an operation, on a function. In most of the cases, the action is nothing more complicated than multiplication or differentiation. Thus, one typical operation might be multiplication by x, which is represented by the operator x x. Another operation might be differentiation with respect to x, represented by the operator d/dx.
It has no physical meaning if written alone. Square root (√) is an operator and is meaningless if written alone but if a quantity or a number is put under it, it transform that quantity or number into its square root, a another quantity or number. Similarly, d/dx is an operator which transforms a function into its first derivative with respect to x.
d/dx (sin x) = cos x
d/dx (xn) = n . xn − 1
Operator . Function = New Function

Linear Operators

An operator is said to be linear if its applications on the some of two functions gives the result which is equal to the operations on the two functions separately.
Ω(aƒ + bg) = aΩƒ + bΩg
where, Ω is operator, a and b are constants and ƒ and g are functions.
d/dx is a linear operator because,
d/dx[ƒ(x) + g(x)] = d/dx(ƒ(x) + d/dx(g(x)

The square root (√) operator is non-linear operator as-
√[ƒ(x) + g(x)] ≠ √ƒ(x) + √g(x)

Commutator Operator

Commutator is a mathematical operation between two operators. For any two operators say Ω and Ω', the difference Ω Ω' − Ω' Ω, which is simply denoted by [Ω,Ω'] is called commutator operator.
If Ω and Ω' commute then [Ω Ω'] = 0, where 0 is called the zero operator which means multiplying a function by zero. If [Ω Ω'] ≠ 0, then two operators are not commute with each other.