Uncertainty Principle - Introduction

Introduction

The uncertainty principle can be interpreted in either the wave mechanics or matrix mechanics formalisms of quantum mechanics. Although the principle is more visually intuitive in the wave mechanics formalism, it was first derived and is more easily generalized in the matrix mechanics formalism. We will attempt to motivate the principle in the two frameworks.

Mathematically, the uncertainty relation between position and momentum arises because the expressions of the wavefunction in the two corresponding bases are Fourier transforms of one another (i.e., position and momentum are conjugate variables). A similar tradeoff between the variances of Fourier conjugates arises wherever Fourier analysis is needed, for example in sound waves. A pure tone is a sharp spike at a single frequency. Its Fourier transform gives the shape of the sound wave in the time domain, which is a completely delocalized sine wave. In quantum mechanics, the two key points are that the position of the particle takes the form of a matter wave, and momentum is its Fourier conjugate, assured by the de Broglie relation, where is the wavenumber.

In the mathematical formulation of quantum mechanics, any pair of non-commuting self-adjoint operators representing observables are subject to similar uncertainty limits. An eigenstate of an observable represents the state of the wavefunction for a certain measurement value (the eigenvalue). For example, if a measurement of an observable is taken then the system is in a particular eigenstate of that observable. The particular eigenstate of the observable may not be an eigenstate of another observable . If this is so, then it does not have a single associated measurement as the system is not in an eigenstate of the observable.