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Linear Harmonic Oscillator Derivation. 718 where nis the vibrational quantum number and q k. Harmonic oscillator has energy levels given by E n n 1 2h n 1 2. This corresponds to the problem in harmonic approximation. Consider a system containing a block of mass m attached to a massless spring with stiffness constant or force constant or spring constant k placed on a smooth horizontal surface frictionless surface as shown in Figure 1013.
BSc III Paper First Physics. 42 It comprises one of the most important examples of elementary Quantum Mechanics. The linear harmonic oscillator describes vibrations in molecules and. This derivation offers the instructor a way to rationalize the energy level spectrum of a harmonic oscillator without having to solve the complicated differential equation prescribed by the Schrodinger equation for the harmonic oscillator. The energy eigenstates of the harmonic oscillator form a family labeled by n coming from Eφˆ x. In this work we present a heuristic derivation for the energy levels of a linear harmonic oscillator that involves only algebra.
There are sev-eral reasons for its pivotal role.
0 1 µn1 µj 2ǫ2ω2 j. The linear harmonic oscillator describes vibrations in molecules and. Linear simple harmonic motion is defined as the motion of a body in which the body performs an oscillatory motion along its path. The force or the acceleration acting on the body is directed towards a fixed point ie. Note that the energy level are equally spaced and the zero-point energy E 0 1 2. The linear harmonic oscillator problem is one of the most fascinating problems in quantum mechanics.