Quantum harmonic oscillator Totally Explained

Everything about The Quantum Harmonic Oscillator totally explained

The quantum harmonic oscillator is the quantum mechanical analogue of the classical harmonic oscillator. It is one of the most important model systems in quantum mechanics because an arbitrary potential can be approximated as a harmonic potential at the vicinity of a stable equilibrium point. Furthermore, it's one of the few quantum mechanical systems for which a simple exact solution is known.

One-dimensional harmonic oscillator

Hamiltonian and energy eigenstates

In the one-dimensional harmonic oscillator problem, a particle of mass

displaystyle m

is subject to a potential

displaystyle V(x) = frac (x_i - x_j)^2.

The potential energy is summed over "nearest-neighbor" pairs, so there's one term for each spring.
Remarkably, there exists a coordinate transformation to turn this problem into a set of independent harmonic oscillators, each of which corresponds to a particular collective distortion of the lattice. These distortions display some particle-like properties, and are called phonons. Phonons occur in the ionic lattices of many solids, and are extremely important for understanding many of the phenomena studied in solid state physics.

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