Editing Adiabatic theorem (section)

=== Quantum harmonic oscillator ===

[[Image:HO adiabatic process.gif|thumb|right|300px|'''Figure 1.''' Change in the probability density, <math>|\psi(t)|^2</math>, of a ground state quantum harmonic oscillator, due to an adiabatic increase in spring constant.]]

The [[Classical physics|classical]] nature of a pendulum precludes a full description of the effects of the adiabatic theorem. As a further example consider a [[quantum harmonic oscillator]] as the [[spring constant]] <math>k</math> is increased. Classically this is equivalent to increasing the stiffness of a spring; quantum-mechanically the effect is a narrowing of the [[potential energy]] curve in the system [[Hamiltonian (quantum mechanics)|Hamiltonian]].

If <math>k</math> is increased adiabatically <math display="inline">\left(\frac{dk}{dt} \to 0\right)</math> then the system at time <math>t</math> will be in an instantaneous eigenstate <math>\psi(t)</math> of the ''current'' Hamiltonian <math>\hat{H}(t)</math>, corresponding to the initial eigenstate of <math>\hat{H}(0)</math>. For the special case of a system like the quantum harmonic oscillator described by a single [[quantum number]], this means the quantum number will remain unchanged. '''Figure 1''' shows how a harmonic oscillator, initially in its ground state, <math>n = 0</math>, remains in the ground state as the potential energy curve is compressed; the functional form of the state adapting to the slowly varying conditions.

For a rapidly increased spring constant, the system undergoes a diabatic process <math display="inline">\left(\frac{dk}{dt} \to \infty\right)</math> in which the system has no time to adapt its functional form to the changing conditions. While the final state must look identical to the initial state <math>\left(|\psi(t)|^2 = |\psi(0)|^2\right)</math> for a process occurring over a vanishing time period, there is no eigenstate of the new Hamiltonian, <math>\hat{H}(t)</math>, that resembles the initial state. The final state is composed of a [[linear superposition]] of many different eigenstates of <math>\hat{H}(t)</math> which sum to reproduce the form of the initial state.