Study Notes On Simple Harmonic Oscillator(Chemical Science) - Download PDF!

By Astha Singh|Updated : May 5th, 2022

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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: