3.2 The Heisenberg Uncertainty Principle

The Heisenberg uncertainty principle is a way of expressing the qualitative properties of quantum mechanics in an easy to visualize way.

Figure 3.3: Illustration of the Heisenberg uncertainty principle. A combination plot of position and linear momentum components in a single direction is shown. Left: Fairly localized state with fairly low linear momentum. Right: narrowing down the position makes the linear momentum explode.
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Figure 3.3 is a combination plot of the position $x$ of a particle and the corresponding linear momentum $p_x$ $\vphantom0\raisebox{1.5pt}{$=$}$ $mv_x$, (with $m$ the mass and $v_x$ the velocity in the $x$-​direction). To the left in the figure, both the position and the linear momentum have some uncertainty.

The right of the figure shows what happens if you squeeze down on the particle to try to restrict it to one position $x$: it stretches out in the momentum direction.

Heisenberg showed that according to quantum mechanics, the area of the blob cannot be contracted to a point. When you try to narrow down the position of a particle, you get into trouble with momentum. Conversely, if you try to pin down a precise momentum, you lose all hold on the position.

The area of the blob has a minimum value below which you cannot go. This minimum area is comparable in size to the so-called Planck constant, roughly 10$\POW9,{-34}$ kg m$\POW9,{2}$/s. That is an extremely small area for macroscopic systems, relatively speaking. But it is big enough to dominate the motion of microscopic systems, like say electrons in atoms.


Key Points
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The Heisenberg uncertainty principle says that there is always a minimum combined uncertainty in position and linear momentum.

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It implies that a particle cannot have a mathematically precise position, because that would require an infinite uncertainty in linear momentum.

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It also implies that a particle cannot have a mathematically precise linear momentum (velocity), since that would imply an infinite uncertainty in position.