Quantum Operators & Functions
OPERATORS
An operator is a symbol that is used to express mathematical calculations/procedures/commands in which one function changes to another one.
Operator (Function) = New Function
Algebra of Operators
1. Addition and Subtraction of Operators: If Aˆ and Bˆ are two different operators and f(x) is the function, then,
2. Multiplication of Operators: If Aˆ and Bˆ are two different operators and f(x) is the function, then order of the operation is very important and it can be represented as:
3. Commutative Property: If two operators are such that the result of their successive applications remains same irrespective of the order of operations, then the two operators are said to be commutative.
EigenFunction, EigenValue and Eigen value Equation: If an operator operates on a well-behaved function f(x), and gives the same function multiplied by a constant. Then, the function f(x) is called the eigen function and the constant is called the eigen value of the operator. The equation formed is called as Eigen Value Equation i.e.,
Some of the Common Operators used in Quantum Mechanics:
1. Laplacian Operator: It is an important differential operator used in quantum mechanics and it can be represented as:
2. Momentum Operator :
3. Square of Momentum Operator (p2):
4.
5.
6.
In Quantum Mechanics We Deal with Only Linear Hermitian Operators:
1. Linear Operators: If it is applied on the sum of two functions, then the result is equal to the sum of the operations on the two functions separately.
Let operator Aˆ operate on two functions of f and g, then,
2. Hermitian Operators: Suppose ψ and φ are the two eigen functions of the operators Aˆ and if
then the operator Aˆ is called the Hermitian operator.
Two properties of Hermitian Operator are:
- Eigen Values of Hermitian Operators are Real (Positive or Negative).
- Eigen Functions of a Hermitian Operator Corresponding to different Eigen values are Orthogonal.
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