Algebraic Solution of the Quantum Harmonic Oscillator

Abstract

The Wave-particle duality and the Schrodinger cat paradox pose important notions on quantum mechanical systems, in which a quantum system can exist in two physically opposite or distinct states. This motivates scientists from the days of Louis de Broglie to confidently look at matter as having a dual nature, which is a physical particle that can also be visualized as a wave entity. This paper seeks to provide an elegant method of solving the time-independent Schrodinger Equation (TISE) for a quantum simple harmonic oscillator (QSHO). A fully algebraic method is presented, the paper explains in detail how simple algebra, coupled with basic quantum mechanical operator commutation algebra can be employed to solve the TISE without resorting to complicated mathematical methods of solving second-order differential equations. The construction of the Hamiltonian is explained and an analogy between the classical simple harmonic oscillator (CSHO) and the QSHO is drawn to develop the solution method. The action of creation and annihilation operators on wave functions is exploited to generate states with higher and lower energies respectively and to generate a Hamiltonian that can act on a system and generate required quantum states. The aim is to generate the ground state of the QSHO and its associated energy value. From this state, other higher energy states can be generated by the action of the creation operator. The use of algebraic methods has been seen to be versatile as it does not explicitly rely on any particular basis or coordinate system. The method allows operators to be manipulated in different bases for example the momentum operator can be given as P ̂ x=-iℏ (∂/∂x) in a spatial coordinate basis or as a pure number p ̂ =p in the momentum basis.