Heisenberg Uncertainty Principle

Once we accept that particles like electrons are also waves, we face a fundamental limit: we cannot simultaneously know both the exact position and exact momentum of a particle.

Proposed by Werner Heisenberg in 1927.

The Core Idea

Why Position and Momentum?

  • A wave is spread out — it doesn't have one exact point
  • If an electron behaves like a wave, it doesn't have a precise location
  • The more we try to "pin down" its position, the less we know about its momentum

Heisenberg's Statement

"The more accurately we know position, the less accurately we know momentum, and vice versa."

The Mathematical Form

$$\Delta x \cdot \Delta p \geq \frac{h}{4\pi}$$

Where:

Symbol Meaning Units
$\Delta x$ Uncertainty in position m
$\Delta p$ Uncertainty in momentum kg·m/s
$h$ Planck's constant $6.63 \times 10^{-34}$ J·s

Minimum Uncertainty

For calculations, we often use the minimum equality: $$\Delta x \cdot \Delta p = \frac{h}{4\pi} = 5.28 \times 10^{-35} \text{ J·s}$$

Physical Interpretation

This is NOT

  • A limitation of our measuring instruments
  • An engineering problem that better technology can fix

This IS

  • A fundamental property of nature
  • A consequence of the wave nature of matter
  • Built into the fabric of quantum mechanics

The Wave Picture

  • A wave with a precise wavelength (definite momentum) extends infinitely — no precise position
  • A wave localized to a point (precise position) requires infinite wavelengths — no precise momentum

Energy-Time Uncertainty

Another form of the principle: $$\Delta E \cdot \Delta t \geq \frac{h}{4\pi}$$

The shorter the lifetime of a state ($\Delta t$), the greater the uncertainty in its energy.

Example Calculations

Example 1: Finding Momentum Uncertainty

An electron's position is measured with uncertainty $\Delta x = 1.0 \times 10^{-10}$ m. Find $\Delta p$.

$$\Delta p = \frac{h}{4\pi \cdot \Delta x} = \frac{5.28 \times 10^{-35}}{1.0 \times 10^{-10}} = 5.28 \times 10^{-25} \text{ kg·m/s}$$

Example 2: Finding Position Uncertainty

An electron has momentum uncertainty $\Delta p = 1.0 \times 10^{-28}$ kg·m/s. Find $\Delta x$.

$$\Delta x = \frac{h}{4\pi \cdot \Delta p} = \frac{5.28 \times 10^{-35}}{1.0 \times 10^{-28}} = 5.28 \times 10^{-7} \text{ m}$$

Example 3: Velocity Uncertainty

An electron is confined in an atom ($\Delta x = 5.0 \times 10^{-11}$ m). Find the minimum velocity uncertainty.

$$\Delta p = \frac{h}{4\pi \cdot \Delta x} = \frac{5.28 \times 10^{-35}}{5.0 \times 10^{-11}} = 1.06 \times 10^{-24} \text{ kg·m/s}$$

$$\Delta v = \frac{\Delta p}{m} = \frac{1.06 \times 10^{-24}}{9.11 \times 10^{-31}} = 1.16 \times 10^6 \text{ m/s}$$

Key Takeaways

  1. Fundamental Limit: Not a measurement error — it's how nature works
  2. Wave Origin: Arises because particles have wave properties
  3. Trade-off: Precise position → fuzzy momentum; precise momentum → fuzzy position
  4. Macroscopic vs Quantum: Uncertainty negligible for everyday objects, dominant at quantum scale

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