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Concept
Quantum harmonic oscillator
Summary
The quantum harmonic oscillator is the quantum-mechanical analog of the classical harmonic oscillator. Because an arbitrary smooth potential can usually be approximated as a harmonic potential at the vicinity of a stable equilibrium point, it is one of the most important model systems in quantum mechanics. Furthermore, it is one of the few quantum-mechanical systems for which an exact, analytical solution is known.
The Hamiltonian of the particle is:
where m is the particle's mass, k is the force constant, is the angular frequency of the oscillator, is the position operator (given by x in the coordinate basis), and is the momentum operator (given by in the coordinate basis). The first term in the Hamiltonian represents the kinetic energy of the particle, and the second term represents its potential energy, as in Hooke's law.
The time-independent Schrödinger equation is,
where E denotes a real number (which needs to be determined) that will specify a time-independent energy level, or eigenvalue, and the solution ψ⟩ denotes that level's energy eigenstate.
Then solve the differential equation representing this eigenvalue problem in the coordinate basis, for the wave function ⟨xψ⟩ = ψ(x), using a spectral method. It turns out that there is a family of solutions. In this basis, they amount to Hermite functions,
The functions Hn are the physicists' Hermite polynomials,
The corresponding energy levels are
This energy spectrum is noteworthy for three reasons. First, the energies are quantized, meaning that only discrete energy values (integer-plus-half multiples of ħω) are possible; this is a general feature of quantum-mechanical systems when a particle is confined. Second, these discrete energy levels are equally spaced, unlike in the Bohr model of the atom, or the particle in a box. Third, the lowest achievable energy (the energy of the n = 0 state, called the ground state) is not equal to the minimum of the potential well, but ħω/2 above it; this is called zero-point energy.
Official source
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