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Before proceeding any further, it is useful to introduce a new coordinate system that is particularly suited to the problems we shall encounter.
We can extend the two dimensional model of the particle on a ring to three dimensions by considering a particle that is constrained to move on the surface of a sphere of radius r. (Clearly this will be useful when it comes to modelling the behaviour of electrons in atoms.)
A particle performs harmonic motion if it experiences a restoring force proportional to its displacement, x, from a given point. i.e. F = -k x. k is called the force constant.
This is a fundamental principle of Quantum mechanics. Loosely speaking, it states that: It is impossible to specify precisely both the linear momentum and the position of a particle at a single moment in time.
In this model, we consider a particle that is confined to a rectangular plane, of length Lx in the x direction and Ly in the y direction. The potential energy is zero everywhere in this plane, and infinite at its walls and beyond. It should be clear that this is an extension of the particle in a one-dimensional box to two dimensions.
We now consider a situation where a particle is confined by walls of potential energy V, the potential being zero within the walls; the potential does not rise to infinity at the walls.
The heading of this page refers to the hypothetical system that will be considered on it. Though in itself this problem may appear quite a trivial one, it introduces various important concepts, and paves the way for exploration of some slightly more complex and physically relevant systems. It may also itself be used as a first approximation to some actual physical problems.