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Quantum Mechanics

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Polar Coordinates

Before proceeding any further, it is useful to introduce a new coordinate system that is particularly suited to the problems we shall encounter.

Rotation in Three Dimensions

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.)

Vibrational Motion

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.

Heisenberg’s Uncertainty Principle

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.

The Particle in A Two-Dimensional Box

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.

Particle Tunnelling

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 Particle in a One-Dimensional Box

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.