Revision Notes for Chemical Science: Quantum Operators & Functions! (Download PDF)

By Renuka Miglani|Updated : December 31st, 2021

Hello Aspirants,

Are you looking for some short and reliable notes during your CSIR-NET preparations? Then, you have come to a perfect place!

Candidates preparing for their CSIR NET exam might need to get some short study notes and strategies to apply while preparing for the key exam of their life. At this point, We at Byjus Exam Prep come up with short notes on the Quantum Operators & Functionswhich comes under the Physical Chemistry section of the Chemical Science syllabus. 

Our experienced subject matter experts have meticulously designed this set of short notes on the Quantum Operators & Functions to give you the most standard set of study materials to be focused upon. In this cut-throat competitive world, students need to prepare themselves with the best study materials to help them learn and for their future. So, here we are offering the best study notes that are reliable and can be used by the students during their preparations for the upcoming CSIR-NET 2021 exam.

Quantum Operators & Functions


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.

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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):









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:

  1. Eigen Values of Hermitian Operators are Real (Positive or Negative). 
  2. Eigen Functions of a Hermitian Operator Corresponding to different Eigen values are Orthogonal.

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