**Classical harmonic oscillator** Classical force on the oscillating bodies:

F = -k.x

__Equations of motion:__ F = -k.x = m.a = m d

^{2}x/dt

^{2} possible solution: x(t) = Acos(2πft+Φ)

Taking the derivatives:

d/dt x(t) = -A2πfsin(2πft+Φ)

d

^{2}/dt

^{2} x(t) = -A(2πf)

^{2}cos(2πft+Φ)

Solving:

-kx = m d

^{2}x/dt

^{2} -kAcos(2πft+Φ) = -mA(2πf)

^{2}cos(2πft+Φ)

k = m(2πf)

^{2} k/m = (2πf)

^{2} √(k/m) = 2πf

ω = √(k/m)

ω = angular frequency (2π.f)

k = force constant, spring stiffness coeff, spring constant

m = reduced mass, moment of inertia

__Potential energy:__ To calculate the potential energy, the energy stored in the spring, consider the spring at rest at x=0. We stretch it from x=0 to x to give it its initial potential energy U:

U =

_{0}∫

^{x} Fdx =

_{0}∫

^{x} kx dx = ½kx

^{2} __Total energy:__ U

_{total} = U

_{kinetic} + U

_{potential} = ½mv

^{2} + ½kx

^{2} Quantum harmonic oscillator:

E(n) = (n+1/2).(h/2pi).√(k/m)

Reference:

http://en.wikipedia.org/wiki/Quantum_harmonic_oscillator __see also:__ Coherent States of the Simple Harmonic Oscillator

http://galileo.phys.virginia.edu/classes/751.mf1i.fall02/CoherentStates.htm