The quantum linear harmonic oscillator, and hence coherent states, arise in the quantum theory of a wide range of physical systems.
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A very important use of complex numbers is for solving the differential equation of a linear harmonic oscillator : d ^ 2y / dx ^ 2 + y = 0.
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The energy eigenstates of the linear harmonic oscillator ( e . g ., masses on springs, lattice vibrations in a solid, vibrational motions of nuclei in molecules, or oscillations in the electromagnetic field ) are fixed-number quantum states.
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