Q.If the uncertainty in the position of an electron is taken to be $1 \times 10^{-15}\ \text{m}$ (of the order of a nuclear diameter), calculate the minimum uncertainty in its velocity, and comment on what the result implies about whether an electron can exist inside the nucleus.

The Heisenberg Uncertainty Principle: Why Nature Won't Let You Be Too Precise

Imagine you're trying to photograph a speeding car at night. To see where it is, you need light — but light carries energy. If you use a dim flashlight, the car barely gets lit, and your photo is blurry. If you use a bright flash, you see the car sharply — but the flash itself pushes the car slightly, changing its speed. You can never know both exactly where it was and exactly how fast it was going after the flash.

That's the intuition. But for an electron, the problem isn't a clumsy flashlight — it's built into the fabric of reality.

The core idea

An electron is not a tiny billiard ball with a definite position and speed. It's a quantum object that behaves like a wave. A wave, by its nature, is spread out — you cannot point to a single point and say "the wave is exactly here." The more you try to pin down where the wave is, the less you know about its wavelength (which tells you its momentum). And the more precisely you measure its momentum, the more the wave spreads out in space.

This trade-off is not a limitation of your instruments. It is a fundamental law of nature: you cannot simultaneously know both the exact position and the exact momentum of a particle.

The precise statement

For any particle, the uncertainties in its position ($\Delta x$) and its momentum ($\Delta p$) satisfy:

where $h$ is Planck's constant ($6.626 \times 10^{-34} \ \text{J·s}$).

What it means for an electron

If you try to measure an electron's position very precisely ($\Delta x$ very small), the uncertainty in its momentum becomes huge ($\Delta p$ very large). The electron's speed becomes wildly unpredictable. Conversely, if you measure its momentum precisely, you lose track of where it is.

Why it's not about measurement

A common misunderstanding: "We can't measure both because the measurement disturbs the particle." That's part of the story, but not the whole truth. Even in principle, with perfect instruments, the uncertainty remains. The electron does not have a simultaneous definite position and momentum — the question itself is meaningless in the quantum world.

The energy-time version

There is also an uncertainty relation between energy and time:

$\Delta E \cdot \Delta t \geq \frac{h}{4\pi}$ …