2.2 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 2.2 is a combination plot of the position $x$ of a particle and the corresponding linear momentum $p_x = m u$, (with $m$ the mass and $u$ the velocity in the $x$-direction):

Figure 2.2: Combined plot of position and momentum components.

Figure 2.3 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:

Figure 2.3: The uncertainty principle illustrated.

Heisenberg showed that according to quantum mechanics, the area of the blue “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.

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.