A quantum-mechanical system that has the properties of a harmonic oscillator, i.e. a restorative force is present that causes the system to oscillate about a point of equilibrium. These systems are described by Schrödinger's Equation with the Hamiltonian of a harmonic oscillator:

     p²      Kx²
H = ---  +  ---
     2m      2
where p and x are the position and momentum operators, K is the force constant of the harmonic oscillator, and m is the mass of the oscillating body. Thus Schrödinger's Equation:
H|E⟩ = E|E⟩

can be cast into position space:

                  
-ħ² d²         mω²   
 -- --- ⟨x|E⟩ + --- x²  ⟨x|E⟩ = E⟨x|E⟩
 2m dx²         2

where ħ is Planck's constant and ω is the natural frequency of oscillation (the square root of the quotient of the force constant of the oscillator and the mass of the oscillating body). This is a differential equation for the energy eigenstate wave functions. Only the solutions to this equation which can correspond to physical solutions are valid, of course, and for the results to be valid, any candidate solutions must certainly be continuous, single-valued, and normalizable (i.e. square-integrable) functions everywhere. We note that this differential equation is isomorphic to the Hermite differential equation, which can be shown by making a few substitutions, thus, the solutions that arise are in the form of Hermite polynomials, if the energies that arise are restricted to E = (n+1/2)ħ;ω. Values for the energy other than these produce solutions which do not satisfy the conditions for valid wave functions (which are confluent hypergeometric functions and are not square-integrable). The energy of such a harmonic oscillator is thus quantized in equal steps of ħω.

It is interesting to note that even at the lowest energy level, with n=0, there is still a zero point energy E0 = (1/2)ħω.

While seldom actually appearing in practice, the quantum harmonic oscillator serves as a useful approximation for many problems when only small vibrations about the equilibrium point occur, because the functions that describe any restoring force in a real-world problem can be approximated by that of a simple harmonic oscillator, provided the displacements are small. A Maclaurin series expansion for such a function should be sufficient to show it.

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