## Uncertainty from another aspect

It is interesting to see how the methods of wave mechanics lead to the uncertainty principle, albeit by a different route to that chosen by Heisenberg:

The matter wave, or wavefunction, of a particle of momentum p is a continuous sinusoidal wave of wavelength X = h/p. The amplitude of the wave (or, more precisely, the square of the amplitude) is a measure of the probability of finding the particle in a given spot, but in the case of a continuous wave the amplitude does not change in space. The chance of finding the particle is the same everywhere.

Amplitude

Position

We know the momentum, but we know nothing about the position.

the uncertainty in position is Ax

A localised particle is represented by a wave packet. This gives the most likely place to find the particle. Since the borders of the wave packet are not sharp, the width, Ax, is approximate (but can be defined more rigorously by a statistical formula).

We have seen in Section 6.8 that a square wave is made up of the superposition of sine waves of different wavelengths. Similarly, a wave packet of matter waves consists of a series of waves of different wavelengths X, and therefore different values of momentum p. We can relate an uncertainty in position to the corresponding uncertainty in momentum using the results of a more exact the uncertainty in position is Ax mathematical calculation, known as Fourier analysis, which states that a combination of sine waves in the wavelength range AX will form a wave packet of width Ax, where

2pp h

(the uncertainty principle)

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