Skip to content

Heisenberg Uncertainty Principle

The Formula

\[ \Delta x \cdot \Delta p \geq \frac{\hbar}{2} \]

More generally, for any two observables \(A\) and \(B\):

\[ \Delta A \cdot \Delta B \geq \frac{1}{2} \left| \langle [\hat{A}, \hat{B}] \rangle \right| \]

Where \([\hat{A}, \hat{B}] = \hat{A}\hat{B} - \hat{B}\hat{A}\) is the commutator.

For energy and time:

\[ \Delta E \cdot \Delta t \geq \frac{\hbar}{2} \]

What It Means

The Heisenberg Uncertainty Principle states that certain pairs of physical properties cannot be simultaneously known with arbitrary precision. The more precisely you know one property, the less precisely you can know the other.

For position and momentum: * \(\Delta x\) is the uncertainty in position * \(\Delta p\) is the uncertainty in momentum * Their product has a fundamental lower bound of \(\frac{\hbar}{2}\)

This is not a limitation of measurement technology — it's a fundamental property of nature. Particles don't have precise positions and momenta simultaneously.

Why It Works — The Intuition

The uncertainty principle arises from the wave nature of matter. Consider a wave:

  • A wave with a perfectly defined wavelength (and thus momentum) extends infinitely in space — its position is completely uncertain.
  • A wave localized at a specific point is a "spike" made of many different wavelengths — its momentum is completely uncertain.

The key insight is the Fourier relationship between position and momentum. In quantum mechanics:

  • Position space and momentum space are Fourier transforms of each other
  • A narrow peak in position space requires many frequencies (momenta) to construct
  • A single frequency (definite momentum) gives a wave spread over all space

Think of it like sound: a pure tone (single frequency) lasts forever, while a short pulse contains many frequencies. You can't have both a perfectly short pulse AND a perfectly pure frequency.

Derivation

From Wave Packets

A quantum particle is described by a wave packet \(\psi(x)\). The spread in position is:

\[ (\Delta x)^2 = \langle x^2 \rangle - \langle x \rangle^2 \]

Similarly for momentum:

\[ (\Delta p)^2 = \langle p^2 \rangle - \langle p \rangle^2 \]

Using the Cauchy-Schwarz Inequality

For any two operators \(\hat{A}\) and \(\hat{B}\), define:

\[ f = (\hat{A} - \langle A \rangle)\psi, \quad g = (\hat{B} - \langle B \rangle)\psi \]

The Cauchy-Schwarz inequality states:

\[ \langle f | f \rangle \langle g | g \rangle \geq |\langle f | g \rangle|^2 \]

This gives:

\[ (\Delta A)^2 (\Delta B)^2 \geq |\langle (\hat{A} - \langle A \rangle)(\hat{B} - \langle B \rangle) \rangle|^2 \]

Separating Real and Imaginary Parts

Any complex number \(z\) satisfies \(|z|^2 \geq (\text{Im}(z))^2\). For the right side:

\[ \langle (\hat{A} - \langle A \rangle)(\hat{B} - \langle B \rangle) \rangle = \frac{1}{2}\langle \{\hat{A}, \hat{B}\} \rangle - \langle A \rangle\langle B \rangle + \frac{1}{2}\langle [\hat{A}, \hat{B}] \rangle \]

where \(\{\hat{A}, \hat{B}\} = \hat{A}\hat{B} + \hat{B}\hat{A}\) is the anticommutator.

The imaginary part comes from the commutator:

\[ |\langle (\hat{A} - \langle A \rangle)(\hat{B} - \langle B \rangle) \rangle|^2 \geq \frac{1}{4} |\langle [\hat{A}, \hat{B}] \rangle|^2 \]

The General Uncertainty Relation

Taking the square root:

\[ \Delta A \cdot \Delta B \geq \frac{1}{2} \left| \langle [\hat{A}, \hat{B}] \rangle \right| \]

Position-Momentum Case

For position and momentum, the canonical commutation relation is:

\[ [\hat{x}, \hat{p}] = i\hbar \]

Substituting:

\[ \Delta x \cdot \Delta p \geq \frac{1}{2} |i\hbar| = \frac{\hbar}{2} \]

Energy-Time Case

The energy-time relation is subtly different. Time is not an operator in standard quantum mechanics, but the relation can be derived from:

\[ \Delta E \cdot \Delta t \geq \frac{\hbar}{2} \]

where \(\Delta t\) represents the characteristic time for a system to change appreciably.

Variables Explained

Symbol Name Description
\(\Delta x\) Position Uncertainty Standard deviation of position measurements
\(\Delta p\) Momentum Uncertainty Standard deviation of momentum measurements
\(\Delta E\) Energy Uncertainty Standard deviation of energy measurements
\(\Delta t\) Time Uncertainty Characteristic time for system evolution
\(\hbar\) Reduced Planck Constant \(h/2\pi \approx 1.055 \times 10^{-34}\) J·s
\([\hat{A}, \hat{B}]\) Commutator \(\hat{A}\hat{B} - \hat{B}\hat{A}\), measures non-commutativity
\(\langle \cdot \rangle\) Expectation Value Average value over many measurements

Worked Example: Electron in an Atom

Problem: Estimate the minimum uncertainty in the velocity of an electron confined to an atom of size \(\Delta x \approx 10^{-10}\) m (1 Ångström).

Step 1: Apply the uncertainty principle.

\[ \Delta x \cdot \Delta p \geq \frac{\hbar}{2} \]

Step 2: Express momentum uncertainty.

\[ \Delta p \geq \frac{\hbar}{2\Delta x} = \frac{1.055 \times 10^{-34}}{2 \times 10^{-10}} = 5.28 \times 10^{-25} \text{ kg·m/s} \]

Step 3: Convert to velocity uncertainty. For an electron with mass \(m_e \approx 9.11 \times 10^{-31}\) kg:

\[ \Delta v = \frac{\Delta p}{m_e} \geq \frac{5.28 \times 10^{-25}}{9.11 \times 10^{-31}} \approx 5.8 \times 10^5 \text{ m/s} \]

Result: The uncertainty in velocity is about \(580\) km/s!

This is why electrons don't "fall into" the nucleus. Confining them to a small space gives them enormous momentum uncertainty, which translates to high kinetic energy. This zero-point energy keeps the atom stable.

Common Mistakes

  • "Measurement disturbs the system": While measurement does disturb quantum systems, the uncertainty principle is deeper — it's about the state itself, not just measurement limitations.
  • "Better instruments will overcome it": No technological improvement can beat the uncertainty principle. It's a law of nature, not an engineering challenge.
  • Confusing \(\Delta x\) with measurement error: \(\Delta x\) is the intrinsic spread in the quantum state, not the precision of your ruler.
  • Applying it to macroscopic objects: For everyday objects, \(\hbar\) is so small that uncertainty is negligible. A 1 kg object with \(\Delta x = 10^{-10}\) m has \(\Delta v \approx 10^{-25}\) m/s.
  • Thinking particles "jitter": The uncertainty isn't due to random motion — it's that position and momentum don't have simultaneous definite values.