Uncertainty Principle

Calculate the minimum uncertainty bound between position and momentum (Δx · Δp ≥ ℏ/2) and explore its physical consequences.

Particle

Input Variable

Mass (m)

1.000e-101.000e+10

Value

1.000e-41000

Scale context

vs. Bohr radius (a₀): ×18.9
vs. atomic size (≈1 Å): ×10.0
vs. nuclear radius (≈1 fm): ×1.00e+6

Ratio of Δx to each reference scale. Values <1 probe sub-structure; values ≫1 indicate a classical regime.

Particle's Compton wavelength (λ_C)

2.43·10^-12 m

When Δx ≲ λ_C, single-particle quantum mechanics breaks down and quantum field theory is required.

Free evolution of the wave packet

Δx·Δp = 5.27e-35 J·s

Time evolutiont/τ = 0.00 · σ(t)/σ₀ = 1.00

t0 s

Position space |ψ(x,t)|²

Δx(t) = ±1.00·10^-9 m

−Δx(t)x₀+Δx(t)

← FOURIER TRANSFORM →

Momentum space |φ(p)|²· time-invariant

Δp = ±5.27·10^-26 kg·m/s

−Δpp₀+Δp

Characteristic time τ

τ = 17.3 fs

A free Gaussian packet spreads as σ(t) = σ₀·√(1+(t/τ)²), while |φ(p)|² is conserved. The product Δx·Δp saturates ℏ/2 only at t=0 and grows afterwards — uncertainty is intrinsic, not a measurement artefact. τ = 2m·(Δx)²/ℏ is the natural spreading scale; we animate in s = t/τ (dimensionless) and display the physical time with SI prefixes.

Results

Minimum Position Uncertainty (Δx)

Minimum Velocity Uncertainty (Δv)

57.88

km/s

Momentum Uncertainty (Δp)

5.27·10^-26

kg·m/s

Energy Uncertainty (ΔE ≈ Δp²/2m)

9.525

meV

Minimum Associated Time (Δt ≥ ℏ/2ΔE)

34.55

de Broglie Wavelength (λ = h/Δp)

12.57

Product Δx·Δp (must saturate ℏ/2)

5.27·10^-35

J·s

Fundamentals & Explanation

The Uncertainty Principle and Kennard's Inequality

Conceptually proposed by Werner Heisenberg in 1927 and formally proven the same year by Earle Hesse Kennard, this principle establishes a fundamental and insurmountable limit to the precision with which certain pairs of conjugate physical properties can be known simultaneously. In 1929, Howard P. Robertson generalized the inequality to any pair of non-commuting observables, which is the form taught in modern quantum mechanics courses.

Epistemological Note:Uncertainty is not a product of imperfect measuring instruments nor the "observer effect" (altering the system by measuring it): it is an intrinsic property of the wave nature of matter and reflects that position and momentum are conjugate variables via the Fourier transform.

Mathematical Formulation

Kennard's inequality for position and momentum follows from the canonical commutator [x̂, p̂] = iℏ. The energy-time relation has a different status: in standard quantum mechanics tis not a Hermitian operator (Pauli's theorem), so Δt is interpreted in the Mandelstam–Tamm sense (1945) as the characteristic time required for an observable to change by one standard deviation.

Position & Momentum

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

Energy & Time

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

The Gaussian wave packet is the only state that saturatesthe inequality (Δx·Δp = ℏ/2 exactly). Any other distribution satisfies Δx·Δp > ℏ/2.

Wave Nature and Wave Packets

In quantum mechanics, a particle with exact linear momentum (Δp = 0) is described as a plane wave extending infinitely through space, leaving its position completely undetermined (Δx → ∞).

To "localize" a particle within a finite region we must superimpose multiple waves of different frequencies (different momenta). This mathematical operation (Fourier Transform) creates a wave packet, but in doing so we inevitably widen the dispersion of momenta that make up the packet. Under free evolution, moreover, the packet spreads in time as σ(t) = σ₀·√(1+(t/τ)²), ceasing to saturate the inequality.

Extreme Physical Limits

Relativistic Limit (Δv → c)

If we attempt to localize a massive particle with extreme precision, Δp grows abruptly. When the velocity dispersion (Δv) exceeds 10% of the speed of light, the classical approximation Δp = m·Δv loses accuracy and must be replaced by Δp = γ·m·Δv with the Lorentz factor.

Compton Limit (ƛ_C)

The absolute limit for localizing a massive particle. If confined to a space smaller than its reduced Compton wavelength (ƛ_C = ℏ/mc), the energy fluctuation ΔE exceeds 2mc² and non-relativistic quantum mechanics ceases to apply: quantum field theory (QFT) is required, where particle-antiparticle pair creation makes the very notion of a localized particle ambiguous.

Typical Scales (Δx)

Macroscopic: > 1 mm

Cell: ~10 μm

Virus: ~100 nm

Atom: ~0.1 nm

Nucleus: ~1 fm

Constants (CODATA)

ℏ ≈ 1.0546 × 10^-34 J·s

h ≈ 6.6261 × 10^-34 J·s

c = 299 792 458 m/s

Academic References

[1] Heisenberg, W. (1927). Über den anschaulichen Inhalt der quantentheoretischen Kinematik und Mechanik. Zeitschrift für Physik, 43, 172–198. doi.org/10.1007/BF01397280
[2] Kennard, E. H. (1927). Zur Quantenmechanik einfacher Bewegungstypen. Zeitschrift für Physik, 44, 326–352. doi.org/10.1007/BF01391200
[3] Robertson, H. P. (1929). The Uncertainty Principle. Physical Review, 34, 163–164. doi.org/10.1103/PhysRev.34.163
[4] Busch, P., Heinonen, T., & Lahti, P. (2007). Heisenberg's uncertainty principle. Physics Reports, 452(6), 155–176. doi.org/10.1016/j.physrep.2007.05.006
[5] Griffiths, D. J., & Schroeter, D. F. (2018). Introduction to Quantum Mechanics (3rd ed.). Cambridge University Press.

Uncertainty Principle

Free evolution of the wave packet

Results

Fundamentals & Explanation

The Uncertainty Principle and Kennard's Inequality

Mathematical Formulation

Wave Nature and Wave Packets

Extreme Physical Limits

Relativistic Limit (Δv → c)

Compton Limit (ƛC)

Typical Scales (Δx)

Constants (CODATA)

Academic References

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The Uncertainty Principle and Kennard's Inequality

Mathematical Formulation

Wave Nature and Wave Packets

Extreme Physical Limits

Relativistic Limit (Δv → c)

Compton Limit (ƛC)

Typical Scales (Δx)

Constants (CODATA)

Academic References

Compton Limit (ƛ_C)

Compton Limit (ƛ_C)