This is an important application of Schrodinger Equation.  I have tried to solve the problem in easy way that can be easily understand by the students of B. Sc. T. Y.

Harmonic motion occurs when system of some kind vibrate about equilibrium position. Ex. Thermal vibration of the atoms in a lattice. Motion of simple pendulum. In simple case of harmonic motion the restoring force on the particle of mass m is linear. This means that restoring force F is directly proportional to displacement of particle from equilibrium position. the problem of harmonic oscillator is conveniently solved by using Schrodinger's equation. I tried here to make the derivation simple for the students of B. Sc. Class.

It is very interesting application  of Schrodinger equation.. we find the solution to Schrodinger equation for particle moving under some type of interaction. the motion of particle in one dimensional box  is to and fro motion in uniform potential. the problem is explore in easy way and better for understanding to the students.

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In many situations the potential energy V is not a function of time and function of position only. Therefore the Schrodinger's equation is modified by removing all references to time explicity. so in this article Schrodinger equation in time dependent form is modified removing all references to time. it is very simple derivation and can easy to understand. 




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The total energy and Momentum are the dynamical quantities and changing with time, therefore difficult to find their expection values hence these quantities are converted into their corresponding differential oprators and used for calculate their expection values.
This article in the B. Sc. Ty syllabus I tried to make simple for the SRTMU B. SC. students.

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The time dependent form of Schrodinger has been derived by using the freely moving particle and not be. interacted. the expression is derived in simple form and easy for understanding for B. Sc. TY students of SRTM university Nanded India.