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An operator is a symbol for a certain mathematical procedure which transforms one function to another.
In other words,
An operator is an instruction to carry out certain operations.
For every physical measurable or observable quantity like position, velocity, linear momentum, angular momentum, energy of the system, there is a corresponding operator in quantum mechanics.
Linear Operator:
While operating on the sum of two functions (f and g), if an operator \((\widehat { A } )\) gives the same result as the sum of the operations on the two functions seperately, then the operator is said to be linear.
Suppose f and g are two functions and \(\widehat { A } \) is an operator then if, $$ \widehat { A } \left( f + g \right) = \widehat { A } f + \widehat { A } g $$ the operator is linear operator.
Vector Operator:
When a vector quantity like momentum is converted into operator, it becomes a vector operator. This vector operator can be obtained in terms of Cartesian Coordinates (x, y and z directions) by following relation. $$ \triangledown = i\frac { \partial }{ \partial x } + j\frac { \partial }{ \partial y } + k\frac { \partial }{ \partial z } $$ where, i, j and k are the unit vectors along the x, y and z coordinates.
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