Heisenberg's Uncertainty Principle

Heisenberg's Uncertainty Principle says the better we know a quantum particle's position the less well we know its momentum, and vice versa.

This is the formula (see other versions):

σ_x σ_p ≥ h4π

Where:

• σ is standard deviation, but let's just call it "uncertainty"
• σ_x is the uncertainty in position
• σ_p is the uncertainty in momentum
• h is Planck's constant, equal to 6.626 070 15 × 10⁻³⁴ Js

It says:

The uncertainty in position times the uncertainty in momentum
is greater than or equal to h4π

• So if σ_x is small, then σ_p must be large
• Or if σ_x is large, then σ_p must be small
• Or any combination like that

Why can't we know both perfectly at the same time?

Quantum particles have properties of both particles and waves, known as wave-particle duality, which is key to the uncertainty principle.

See for yourself. Here is the wave pattern of an electron in a hydrogen atom:

 
Quantum Microscope Image of a Hydrogen Atom
https://dx.doi.org/10.1103/PhysRevLett.110.213001

Waves

We can think about wavelength instead of momentum, as they are linked.

So we can say:

The better we know wavelength the less well we know position, and vice versa.

Here is a simple sine wave:

• We know its wavelength perfectly
• But what is its position?
  

Here is a different wave:

It has a better defined center, but is still spread out a lot.

What is it's wavelength?

It is actually made of these three cosine waves added together:

The result is:

• We know it's position better (but not perfectly)
• But we know its wavelength less well

Here is a similar graph made of 50 different wavelengths:

• We know it's position very well now
• But it has 50 different wavelengths!

So we have learned a simple fact:

The better we know position the less well we know wavelength and vice versa.

And because wavelength and momentum are linked we can also say

The better we know position the less well we know momentum and vice versa.

It is a simple fact about waves.

Simply put: focusing on one aspect of a wave blurs our understanding of the other.

Note: in "Classical Physics" (which we use for larger things like billiard balls, projectiles, planets, etc) this effect is so small that we assume we know both position and momentum well at the same time.

Other Versions

You may see this version:

σ_x σ_p ≥ ħ2

• ħ (h with a bar through it) is reduced Planck's constant, equal to h2π

You may also see people use Δx or Δp instead of σ_x and σ_p , but  Δ is usually the symbol for "change in" not standard deviation, so best avoid it.

Energy and Time

A similar relation exists between energy and lifetime (the duration a quantum state exists before it changes):

σ_E σ_t ≥ ħ2

Where:

• σ_E is the uncertainty in energy
• σ_t is the uncertainty in lifetime of the particle

It means that the shorter the lifetime of a quantum particle the less well defined its energy is. A long-lived particle can have a much better defined energy.

Photons Through a Slit

Let us see another aspect of Heisenberg's Uncertainty Principle

When we shine a laser beam through a thin slit it makes a nice pattern on the wall:

This is due to the wave-like nature of photons (and electrons, etc).

Conclusion

The uncertainty principle is essential to quantum mechanics because it highlights nature's inherent limits. Unlike classical physics, where objects have exact positions and velocities, quantum particles only gain precise values when measured.

This principle explains phenomena like electron clouds in atoms, where particles exist as probabilities rather than fixed orbits. It also underpins technologies like quantum computing and advanced microscopes, which rely on these unpredictable behaviors.