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 singular operator.
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\ħ\frac\partial{\partial x}` ........(1)
and can be generalized in three dimension
`\hat P=-i\ħ\nabla` ........(2)
(ii) Hamiltonian operator
The Hamiltonian operator corresponds to the total energy of the system
`\hat H=-\frac{\ħ^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{\ħ^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{ħ^2}{2m}\frac{\partial^2}{\partial x^2}` ......(2)
and can be generalized in three dimension
`\hat{K.E.}=-\frac{ħ^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{iħ}m\nabla` .........(2)
(v) Angular Operator
Angular momentum requires a more complex discussion but is the cross product of the position operator rand the momentum operator P
`\hat L=-iħ(r\times\nabla)` .........(1)
(vi) Energy Operator
The energy operator from the time dependent Schrodinger equation is
`\hat E=i\ħ\frac\partial{\partial t}` ..........(1)
