It is spread out in space, a perfect sine wave repeating itself indefinitely with a precisely defined constant wavelength. If this is an electron wave, it represents a particle moving along at constant speed, with a well-defined momentum given by the de Broglie equation: p = h/λ. Where is the electron? It is everywhere—or it is equally likely to be anywhere (along the infinite dimension of the wave).1 Its uncertainty of position is infinite; its uncertainty of momentum (and of wavelength) is zero. This is an extreme situation that is at least consistent with the uncertainty principle. If the product of two uncertainties is constant, one of the two can become vanishingly small only if the other becomes infinitely great.
Consider now the result of superposing several wavelengths. The superposition of two waves differing by 10% in wavelength is shown below.
Five cycles from the point of maximum reinforcement, the waves interfere destructively. Five cycles further on, they again reinforce. This alternation of constructive and destructive interference produces partial localization of the wave, a bunching together of the wave into regions each about 10 wavelengths in extent. There is already an important hint about the uncertainty principle in this result. Partial localization has required a mixing of wavelengths, which in turn implies an uncertainty in momentum.
Greater localization is achieved by superposing a large number of different wavelengths. The figure below shows the result of combining 6 different pure waves, whose wavelengths differ successively by 2%.
The total spread of wavelengths remains 10%, or 5% on either side of the average wavelength. We can therefore assign an uncertainty of momentum of about 5% of the dominant momentum:
Δp ≅ 0.05p.
Here p represents an average momentum, related to an average wavelength λ by p = h/λ. In the figure, the six-fold superposition produces a better localization of the wave. (If all possible wavelengths within the given span were superposed, the wave could be completely localized. Outside of a given region of space its amplitude would fall to zero, never to rise again.) The uncertainty of position is roughly the extent of the wave on either side of its center—about five wavelengths in the figure:
Δx ≅ 5λ .
The product of position and momentum uncertainties is, in this example,
ΔxΔp ≅ 0.25λp.
Since the product λp on the right is equal to Planck’s constant h, this equation can be written
ΔxΔp ≅ 0.25h.
Because we have taken rough-and-ready estimates of the uncertainties, the numerical factor 0.25 in this equation has no special significance. The technical definition of uncertainty used in quantum mechanics makes the right side of the uncertainty equation turn out to be exactly h/2π, or ħ, as stated in Essay Q6.
Since a 10% spread in momentum “crushed” the wave packet from infinite extent to a little more than 10 wavelengths, it is a reasonable guess that momenta varying by nearly 100% would be required to narrow the packet down to a single cycle. This is true. Maximum localization, Δx ≅ λ/2, requires that wavelengths (and momenta) differing by as much as a factor of two be superposed. Then Δp ≅ p/2. In these equations, p and λ refer to average values of momentum and wavelength, since neither of these variables is well defined for a localized wave. Again the product ΔxΔp is roughly the same as the product λp, which in turn is equal to Planck’s constant.
A real three-dimensional wave function, as opposed to the hypothetical one-dimensional waves discussed above, is found within the hydrogen atom. Its amplitude, along a radial line drawn across a diameter of the atom, is shown below.
This electron wave function shows a highly localized wave, completing only one cycle of oscillation from one side of the atom to the other. The shape of this wave function, with its central peak, differs markedly from the shape of a pure sine wave. Any shape can be “fashioned” with the right combination of infinitely many wavelengths.
This discussion of wave superposition demonstrates the essential point that the Heisenberg uncertainty principle is a consequence of the wave nature of particles. It is no more profound, and in this form says no more and no less than the de Broglie equation giving the wavelength of material particles. Uncertainty of measurement arises essentially from the nonlocalizability of waves.
1 At an instant of time, the square of such a perfect sine wave has hills and valleys, so that the particle has greater probability to be in some places than other places. If it is a propagating wave, the regions of high and low probability change with time, resulting in an equal average probability for the particle to be found anywhere along the wave.