Uncertainty principle

If the electron is also a wave, the classical notions of “well-defined position” and “well-defined momentum” lose their meaning. A pure wave — a sine wave — has a perfectly defined wavelength $\lambda$, and therefore an exact momentum $p = h/\lambda$; but precisely for this reason it does not have a position: it extends everywhere. To localise a particle in space one must sum many waves with different $\lambda$, building a wave packet; but then $\lambda$ — and with it $p$ — is no longer a single number. Precision in position and precision in momentum are two requirements competing for the same wave: gaining in one means losing in the other.

Heisenberg's uncertainty principle

For a particle it is impossible to know simultaneously with arbitrary precision the position $x$ and the momentum $p$: $\ev{\Delta x \cdot \Delta p \geq \frac{\hbar}{2}}$ where $\Delta x$ and $\Delta p$ are the uncertainties on the values of $x$ and $p$.

It is not an instrumental limitation!

The principle does not say “our instruments are not sensitive enough”. It says that nature itself does not “have” more precise answers: in quantum mechanics a simultaneously exact value of position and momentum simply does not exist. The point is not the observer’s ignorance, but the structure of the world.

A semi-quantitative argument

The de Broglie wave of a particle has $\lambda = h/p$. To “see” the position with precision $\Delta x$ a light wave of wavelength $\lambda_\text{luce} \lesssim \Delta x$ is needed. But this light has photons of momentum $p_\text{luce} = h/\lambda_\text{luce} \geq h/\Delta x$, and it partly transfers this to the particle, disturbing it. The more precise the measurement of $x$, the more violent the disturbance to $p$. The product $\Delta x \cdot \Delta p$ always remains $\geq \hbar/2$.

Generalisations and consequences

The principle does not concern only the position–momentum pair: it extends to other pairs of “conjugate” observables. The most famous links energy and time, $\Delta E \cdot \Delta t \geq \frac{\hbar}{2},$ but it also holds, for example, between different components of angular momentum ($\Delta L_x \cdot \Delta L_y$).

From this inequality arise spectacular consequences:

• Stability of the atom. An electron that “fell” onto the nucleus would have a very small $\Delta x$, hence an enormous $\Delta p$, hence enormous kinetic energy: it would “bounce” back outwards. It is the uncertainty that prevents the collapse of the atom and fixes its dimensions.
• Tunnel effect. The energy–time uncertainty allows a particle to cross barriers that, in classical physics, would be insurmountable — a phenomenon underlying nuclear decay and modern scanning microscopes.

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Topics: Fisica quantistica Concepts: Principio di indeterminazione · Dualismo onda-particella

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