An oscillator is any device that produces periodic variations in the value of some quantity. If these periodic variations can be described by a sinusoidal function, we call it a harmonic oscillator. Often, when we study such systems, we think of a mechanical device where an object is periodically displaced along a specific direction in space. In such a case, the classical mechanics of the system is described by a periodic variation in both the position and the momentum of the object.
At some point, somebody thought it may be a good idea to replace the oscillating object in such a mechanical oscillator by the wave function of a quantum particle. The resulting analysis led to the formulation of a simple quantum Hamiltonian in terms of position and momentum operators. These operators can be replaced by ladder operators (creation and annihilation operators), which gives the basic form of the Hamiltonian of all free field theories: theories without interactions. The interactions are added to this Hamiltonian as additional terms. In view of this role, it is generally considered that the quantum harmonics oscillator is probably the single most important system that has ever been analyzed.
The role that the formalism, which is derived from this analysis, plays in our models of physical quantum systems is not to be contested, but some of the concepts that follow from this analysis may be leading us by our noses. Among the conclusions that are derived from the quantum harmonic oscillator analysis there are some that are questionable.
What is it that gives the quantum harmonic oscillator a quantum characteristic? Is it the fact that the particle is replaced by a wave function? Formally such a wave function is just a normalized continuous function. Although it is interpreted as a quantum wave function, such a function can equally well represent a classical field, if we replace the requirement for the normalization with a requirement for a finite energy.
Or is it because Planck’s constant makes an appearance in the expression? Well, if the variables are suitably redefined in terms of dimensionless variables, Planck’s constant would disappear from the equation. The resulting equation still describes the same physical system. Therefore, Planck’s constant does not play a physically significant role in it. Such an equation can also describe the dynamics in classical scenarios. One such classical scenario is the propagation of light in a gradient index (GRIN) lens medium.
Perhaps it is because the Schrödinger equation for the quantum harmonic oscillator has discrete solutions. Such discrete solutions are interpreted as the discrete energy levels representing the different quanta. Well, in those classical scenarios where the same equation is applicable, such as the GRIN lens, the same discrete solutions exit, but without any associated quantum interpretation. Moreover, the Schrödinger equation for the quantum harmonic oscillator also admits a continuous solution with continuous free parameters representing the oscillating motion of the field.
What are we to conclude now? What value can we derive from the analysis of the quantum harmonic oscillator? Any interpretation of the proposed quantum nature of the system seems to be at best misleading. Although there may exist physical quantum systems that are described by such a quantum harmonic oscillator equation, the same formal mathematical expression is equally applicable in classical systems once Planck’s constant is removed.
In our quest to understand quantum physics, it is important to understand what is not quantum physics. It is also important to separate physics from formalism and not confuse the two.
Flippiefanus 