# Operator in Quantum Mechanics (Linear, Identity, Null, Inverse, Momentum, Hamiltonian, Kinetic Energy Operator...)

## Operator

An * operator* is a mathematical rule that transform a given function into another function. As

`\hat A \| a> = \| b>`

Where the operator 'A' transforms the vector |a > to another vector |b >.

The example of operators are addition, subtraction, multiplication, division, differentiation, integration, operation of grad, div, and curl etc.

## Types of Operator

### (i) Linear Operator

An * operator* P is said to be linear operator if it satisfies the following conditions,

`\hat A(u+v)=\hat Au+\hat Av`

`\hat Ac=c\hat A`

Where **u** and **v** are arbitrary functions c is an arbitrary constant.

**For antilinear operator**

`\hat A(u+v)=\hat A^\ast u+\hat A^\ast v`

### (ii) Identity Operator

The * identity operator* I is an operator, which operating on a function, leaves the function unchanged i.e.

`\hat I\ | a> = \ | a>`

### (iii) Null Operator

The * null operator* is an operator which operating on a function, annihilates the function. Thus if

`\hat O\ | a> = 0`

the **O** is the null operator.

### (iv) Inverse operator

Consider **u** and **v** are two vectors of spaces and a linear operator A relates them by the equation

`\ v=\ u\hat A` .......(1)

If there exist an operator B, which reverse the action of A, such that

`\hat Bv=u`

Then the operator B(=A^{-1}) is called the * inverse of operator* A.

`\hat A\hat B=\hat B\hat A=\hat I`

### (v) Singular and non singular operator

An operator, for which inverse exist, is called * non-singular*. In an operator which has no reciprocal is called a

*operator.*

**singular**## Other commonly used operators

All operators in **quantum mechanics** have eigen function and eigen values. Let us derive some important operators that are valid not only for free particles but also for the bound states.

### (i) Momentum operator

The linear * momentum operator* of particle moving in one dimension is

`\hat P_x=-i\Ä§\frac\partial{\partial x}` ........(1)

and can be generalized in three dimension

`\hat P=-i\Ä§\nabla` ........(2)

### (ii) Hamiltonian operator

The * Hamiltonian operator* corresponds to the total energy of the system

`\hat H=-\frac{\Ä§^2}{2m}\frac{\partial^2}{\partial x^2}+V_{(x)}` ......(1)

and it represents the total energy of the particle of mass m in the potential V_{(x).} The Hamiltonian in three dimensions is

`\hat H=-\frac{\Ä§^2}{2m}\nabla^2+V_{(r)}` ......(2)

### (iii) Kinetic Energy Operator

Classically, the kinetic energy of a particle moving in one dimension in terms of momentum is

`K.E.=\frac{P_x^2}{2m}` .......(1)

Quantum mechanically, the corresponding kinetic ** energy operator** is

`\hat{K.E.}=-\frac{Ä§^2}{2m}\frac{\partial^2}{\partial x^2}` ......(2)

and can be generalized in three dimension

`\hat{K.E.}=-\frac{Ä§^2}{2m}\nabla^2`

### (iv) Velocity Operator

Classically, the linear momentum of a particle of mass m and velocity v is given by

**P** = m **v**.........(1)

Quantum mechanically, the corresponding velocity operator is,

`\hat v=\frac{iÄ§}m\nabla` .........(2)

### (v) Angular Operator

Angular momentum requires a more complex discussion but is the cross product of the position operator **r**and the momentum operator P

`\hat L=-iÄ§(r\times\nabla)` .........(1)

### (vi) Energy Operator

The * energy operator* from the time dependent Schrodinger equation is

`\hat E=i\Ä§\frac\partial{\partial t}` ..........(1)