Quantum mechanics MOC

Uncertainty principle

For any observables and ,

which is the generalized uncertainty principle.

Proof of inequality

Let and be Hermitian operators corresponding to observables. Define

so that and . Then by the Cauchy-Schwarz inequality

Now

and by similar calculation Hence

as claimed.

Minimum uncertainty wavepacket

In order for the inequalities above to become equalities, we require and , implying . Thus a minimum uncertainty has

with real. For and in 1D position-space we get a Gaußian¹

in which case . Hence the uncertainty principle is correct, giving the greatest lower bound on the uncertainty of arbitrary observables.

Intuition: The uncertainty principle for classical waves

Consider a classical wave, e.g. a sinusoidal wave traveling down a long rope. It is difficult to say where the wave is, since it is distributed throughout the rope, but it is possible to give the wavelength. Now consider a localized bump travelling down the rope. In this case, it is possible to describe the position of the bump, but giving its wavelength is difficult, since it has no periodicity. Hence we see that the more precise a wave’s position is, the less precise its wavelength, and vice versa. In quantum mechanics, wavelengths correspond to momenta.¹

Heisenberg uncertainty princple

The Heisenberg uncertainty principle is the special case of

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Footnotes

1. 2018. Introduction to Quantum Mechanics, §3.5.2, pp. 108–109 ↩ ↩²