Heisenberg Uncertainty Principle
The Heisenberg Uncertainty Principle states that it is fundamentally impossible to simultaneously determine both the exact position and exact momentum of a quantum particle with arbitrary precision; the more precisely one is known, the less precisely the other can be known. This is not a limitation of measurement instruments but an intrinsic property of quantum systems arising from the wave nature of matter. A complementary relation exists between energy and time, and the principle has profound implications for atomic stability, electron orbitals, and the zero-point energy of quantum systems.
Key Formula
Δx × Δp ≥ ℏ/2
LaTeX: \Delta x \cdot \Delta p \geq \frac{\hbar}{2}
| Symbol | Meaning | Unit |
|---|---|---|
| Δx | Uncertainty in position | Metres (m) |
| Δp | Uncertainty in momentum | kg·m/s |
| ℏ | Reduced Planck's constant (h/2π = 1.055 × 10⁻³⁴) | J·s |
Worked Example
Problem
An electron's position is known to within Δx = 1.0 × 10⁻¹⁰ m (approximately one atomic diameter). What is the minimum uncertainty in its momentum?
Solution
Step 1: Write the uncertainty relation. Δx · Δp ≥ ℏ/2 Step 2: Solve for minimum Δp. Δp ≥ ℏ / (2 × Δx) Step 3: Substitute values. ℏ = 1.055 × 10⁻³⁴ J·s Δx = 1.0 × 10⁻¹⁰ m Δp ≥ 1.055 × 10⁻³⁴ / (2 × 1.0 × 10⁻¹⁰) Δp ≥ 1.055 × 10⁻³⁴ / 2.0 × 10⁻¹⁰ Δp ≥ 5.275 × 10⁻²⁵ kg·m/s
Answer
Δp ≥ 5.28 × 10⁻²⁵ kg·m/s (minimum momentum uncertainty)
Complementary Variable Pairs in Heisenberg Uncertainty Relations
| Variable 1 | Variable 2 | Relation | Physical Significance |
|---|---|---|---|
| Position (x) | Momentum (p) | Δx·Δp ≥ ℏ/2 | Localisation vs motion knowledge |
| Energy (E) | Time (t) | ΔE·Δt ≥ ℏ/2 | Excited state lifetime and linewidth |
| Angle (φ) | Angular momentum (L) | Δφ·ΔL ≥ ℏ/2 | Rotation and spin uncertainty |
| Electric field | Magnetic field | Related via Maxwell | Vacuum fluctuations |
Interactive Tools
PhET Quantum Mechanics Uncertainty
Visualise position and momentum uncertainties in quantum bound states
Open ToolWolframAlpha Uncertainty Principle
Explore the mathematical form and calculations of the uncertainty principle
Open ToolBrilliant – Uncertainty Principle
Detailed wiki and worked problems on Heisenberg's uncertainty principle
Open Tool
Wikimedia Commons, CC BY-SA
Related Terms
Wave-Particle Duality
Wave-particle duality is the quantum mechanical principle stating that every quantum entity, such as an electron or photon, exhibits both wave-like and particle-like properties depending on how it is observed or measured. In experiments such as the double-slit experiment, particles produce interference patterns characteristic of waves when not observed, but behave as localized particles when detected at specific positions. This duality is central to quantum mechanics and demonstrates that classical concepts of "wave" and "particle" are complementary rather than contradictory descriptions of quantum objects.
Schrödinger Equation
The Schrödinger equation is the fundamental equation of motion in non-relativistic quantum mechanics, describing how the quantum state (wave function) of a physical system evolves over time. Its time-independent form is used to find the allowed energy levels and stationary states of quantum systems such as atoms and molecules. Solutions to the Schrödinger equation yield wave functions from which all measurable properties of a quantum system, including energy eigenvalues, transition probabilities, and electron densities, can be derived.
Wave Function
The wave function (denoted Ψ) is a mathematical function in quantum mechanics that completely describes the quantum state of a particle or system. Its squared modulus |Ψ|² gives the probability density for finding the particle at a given position and time, as interpreted by Max Born in 1926. The wave function must be continuous, single-valued, and square-integrable (normalised so that the total probability integrates to one), and it evolves deterministically according to the Schrödinger equation.
Named after German physicist Werner Heisenberg (1901–1976), who formulated the principle in 1927. "Uncertainty" comes from Latin "incertus" (not certain). Heisenberg was awarded the Nobel Prize in Physics in 1932 for the creation of quantum mechanics.