By Astha Singh|Updated : May 5th, 2022

Are you an Aspirant of CSIR-NET and looking for some short and reliable notes for Chemical Sciences to strong your base for preparations? We have got you covered!

Candidates preparing for their CSIR NET exam can really make their preparation journey easier with the help of some reliable study notes that cover the topics in the most simple way. We at BYJU'S Exam Prep have come up with the idea of providing short notes On Simple Harmonic Oscillator, which comes under the Physical Chemistry section of the Chemical Science syllabus.

The short notes on Simple Harmonic Oscillator are developed by our experienced subject-matter experts to provide you with the most standard and authentic set of study materials to be focused upon. The students need the best resources for their preparation to clear the CSIR NET examination, Here are the most reliable study notes to make the topics easier for you and also help you to save your time for the preparations for the upcoming CSIR-NET 2022 exam.

## Study Notes On Simple Harmonic Oscillator

One-Dimensional Simple Harmonic Oscillator:

A diatomic vibration molecule can be represented with the help of a simple model which is known as a simple harmonic oscillator (S.H.O). The force that acts on the molecule is given by:

f=kx

Here, x is the displacement from the equilibrium position and k is the force constant.

The expression of potential energy V(x) is given by:

Here, m1 and m2 are the atomic masses of the two atoms.

By using the expression of potential energy given in Equation (1), for one-dimensional S.H.O., the Schrodinger equation can be represented as:

Equation (11) is identical in form to a well-known second-order differential equation, called the Hermite equation, that is:

The Hermite equation has solutions which depend upon the value of n.

The Hermite Polynomial having degree n can be defined as:

The energy of S.H.O.: On comparing equations (11) and (12), the energy of the S.H.O. can be obtained.

It has been found out that:

α/β = 2n + 1 …(15)

Substituting for α and β Equation (7),

Or …(17)

From Equation (2),

(k /m)1/2 = 2π …(18)

Combining Equations (17) and (18),

This energy is called the zero-point energy of the oscillator. Classical mechanics predicts that the zero-point energy of the oscillator is zero while quantum mechanics predicts that the zero-point energy is non-zero.

The occurrence of the zero-point energy is consistent with the Heisenberg uncertainty principle.

Graph of the wave function and probability:

Degeneracy for a 2-D harmonic oscillator:

The energy, E = Ex + Ey

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