What is the wave function of a harmonic oscillator?

Submicroscopic harmonic oscillators are popular quantum physics problems because harmonic oscillators are relatively simple systems — the force that keeps a particle bound here is proportional to the distance that the particle is from the equilibrium point.

What is the potential of a harmonic oscillator?

A harmonic oscillator (quantum or classical) is a particle in a potential energy well given by V(x)=½kx². k is called the force constant. It can be seen as the motion of a small mass attached to a string, or a particle oscillating in a well shaped as a parabola.

What is quantum harmonic oscillator used for?

The quantum harmonic oscillator has implications far beyond the simple diatomic molecule. It is the foundation for the understanding of complex modes of vibration in larger molecules, the motion of atoms in a solid lattice, the theory of heat capacity, etc.

Why is the quantum harmonic oscillator important?

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.

Which is Schrodinger’s equation for the simple harmonic oscillator?

From the classical expression for total energy given above, the Schrödinger equation for the quantum oscillator follows in standard fashion: −ℏ22md2ψ(x)dx2+12mω2×2ψ(x)=Eψ(x).

What is the ground state of harmonic oscillator?

The ground state energy is larger than zero. This means that, unlike a classical oscillator, a quantum oscillator is never at rest, even at the bottom of a potential well, and undergoes quantum fluctuations.

What is ground state of harmonic oscillator?

This is the smallest energy allowed by the uncertainty principle. This is a very significant physical result because it tells us that the energy of a system described by a harmonic oscillator potential cannot have zero energy.

What is classical forbidden region?

In a classically forbidden region, the energy of the quantum particle is less than the potential energy so that the quantum wave function cannot penetrate the forbidden region unless its dimension is smaller than the decay length of the quantum wave function.