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Heisenberg Uncertainty Principle Calculator

Calculate position-momentum and energy-time uncertainty relations

Parameters

mⓘ
kg·m/sⓘ
Jⓘ
sⓘ
Show Trail

Controls

xⓘ

Calculated Values

Position-Momentum Product:
0.00;J⋅s0.00;J·s
Energy-Time Product:
0.00;J⋅s0.00;J·s
Minimum Position Uncertainty:
0.00;m0.00;m
Minimum Momentum Uncertainty:
0.00;kg⋅m/s0.00;kg·m/s
Energy Uncertainty (eV):
0.06;eV0.06;eV
Velocity Uncertainty:
109776.91;m/s109776.91;m/s
Minimum Energy Uncertainty:
0.00;J0.00;J
Minimum Time Uncertainty:
0.00;s0.00;s

Examples

Example 1: Electron in Atom

An electron with position uncertainty of 1 Å (10⁻¹⁰ m).

  • Position-Momentum Product: 0.000.00
  • Minimum Position Uncertainty: 0.000.00
  • Velocity Uncertainty: 579000.00579000.00
  • Energy Uncertainty (eV): 0.060.06

Example 2: Quantum Particle

A particle with momentum uncertainty of 1e-24 kg·m/s.

  • Position-Momentum Product: 0.000.00
  • Minimum Momentum Uncertainty: 0.000.00
  • Velocity Uncertainty: 1100000.001100000.00
  • Energy Uncertainty (eV): 0.010.01

Example 3: Short-Lived Particle

A particle with lifetime uncertainty of 1e-15 s.

  • Energy-Time Product: 0.000.00
  • Minimum Energy Uncertainty: 0.000.00
  • Energy Uncertainty (eV): 0.330.33
  • Minimum Time Uncertainty: 0.000.00

Visualization

Heisenberg Uncertainty Principle

The Heisenberg Uncertainty Principle is a fundamental concept in quantum mechanics that states there are inherent limits to the precision with which certain pairs of physical properties can be simultaneously known. This principle was formulated by Werner Heisenberg in 1927 and is one of the cornerstones of quantum theory.

The position-momentum uncertainty relation is: ΔxΔp ≥ ℏ/2, where Δx is the uncertainty in position, Δp is the uncertainty in momentum, and ℏ is the reduced Planck constant. This means that the more precisely we know a particle's position, the less precisely we can know its momentum, and vice versa.

The energy-time uncertainty relation is: ΔEΔt ≥ ℏ/2, where ΔE is the uncertainty in energy and Δt is the uncertainty in time. This relation has important implications for particle physics, quantum tunneling, and the stability of quantum states.

The uncertainty principle arises from the wave-like nature of matter. When we try to localize a particle (reduce position uncertainty), we must use shorter wavelength probes, which impart more momentum to the particle, increasing momentum uncertainty. This is not a limitation of our measuring instruments but a fundamental property of nature.

The uncertainty principle has profound implications for our understanding of reality. It shows that at the quantum level, we cannot have complete knowledge of a system's state, and that observation fundamentally affects the system being observed. This principle is essential for understanding quantum tunneling, zero-point energy, and the stability of atoms.

Key Concepts

  • Position-Momentum Uncertainty: ΔxΔp ≥ ℏ/2
  • Energy-Time Uncertainty: ΔEΔt ≥ ℏ/2
  • Reduced Planck Constant: ℏ = h/2π ≈ 1.055×10⁻³⁴ J·s
  • Wave-Particle Duality: Fundamental cause of uncertainty
  • Measurement Effect: Observation affects quantum systems
  • Minimum Uncertainty: ℏ/2 is the fundamental limit

Real-World Applications

  • Quantum Tunneling: Understanding barrier penetration
  • Atomic Stability: Explaining why electrons don't fall into nucleus
  • Particle Physics: Short-lived particle properties
  • Quantum Computing: Qubit coherence times
  • Spectroscopy: Line broadening and energy resolution

Explore Further

More quantum mechanics tools

  • Schrödinger Equation

    Solve the time-dependent and time-independent Schrödinger equations for quantum systems.

  • Quantum Harmonic Oscillator

    Calculate energy levels, wavefunctions, and quantum properties of harmonic oscillators.

  • Particle in a Box

    Calculate energy levels, wavefunctions, and quantum properties of particles confined in potential wells.

  • Quantum Tunneling

    Calculate tunneling probabilities and transmission coefficients for quantum particles.

  • Quantum Spin

    Calculate spin angular momentum and magnetic moments in quantum systems.

  • Quantum Entanglement

    Analyze entangled states and quantum correlations.

Physics Equations

Position-Momentum Uncertainty:
ΔxΔp≥ℏ2\Delta x \Delta p \geq \frac{\hbar}{2}
Energy-Time Uncertainty:
ΔEΔt≥ℏ2\Delta E \Delta t \geq \frac{\hbar}{2}
Reduced Planck Constant:
ℏ=h2π\hbar = \frac{h}{2\pi}
Minimum Uncertainty Product:
ΔxΔp=ℏ2\Delta x \Delta p = \frac{\hbar}{2}
Velocity Uncertainty:
Δv=Δpm\Delta v = \frac{\Delta p}{m}

Step-by-Step Solution

See how the main results are calculated.

1

Step 1: Calculate Position-Momentum Uncertainty Product

First, we calculate the product of position and momentum uncertainties:

Equation:

ΔxΔp\Delta x \Delta p

Calculation:

ΔxΔp=1.00e−10×1.00e−25=1.00e−35 J\cdotps\Delta x \Delta p = 1.00e-10 \times 1.00e-25 = 1.00e-35 \text{ J·s}

Explanation:

This product must satisfy the uncertainty principle.

2

Step 2: Check Uncertainty Principle

We verify that the uncertainty principle is satisfied:

Equation:

ΔxΔp≥ℏ2\Delta x \Delta p \geq \frac{\hbar}{2}

Calculation:

1.00e−35≥5.27e−351.00e-35 \geq 5.27e-35

Explanation:

The uncertainty principle requires this inequality to hold.

3

Step 3: Calculate Minimum Position Uncertainty

For the given momentum uncertainty, we find the minimum position uncertainty:

Equation:

Δxmin=ℏ2Δp\Delta x_{min} = \frac{\hbar}{2\Delta p}

Calculation:

Δxmin=1.05e−342×1.00e−25=5.27e−10 m\Delta x_{min} = \frac{1.05e-34}{2 \times 1.00e-25} = 5.27e-10 \text{ m}

Explanation:

This is the minimum position uncertainty possible for this momentum uncertainty.

4

Step 4: Calculate Energy-Time Uncertainty Product

We calculate the product of energy and time uncertainties:

Equation:

ΔEΔt\Delta E \Delta t

Calculation:

ΔEΔt=1.00e−20×1.00e−12=1.00e−32 J\cdotps\Delta E \Delta t = 1.00e-20 \times 1.00e-12 = 1.00e-32 \text{ J·s}

Explanation:

This product must also satisfy the energy-time uncertainty relation.

5

Step 5: Convert Energy to eV

We convert the energy uncertainty to electron volts:

Equation:

ΔEeV=ΔEJe\Delta E_{eV} = \frac{\Delta E_J}{e}

Calculation:

ΔEeV=1.00e−201.60e−19=0.0624 eV\Delta E_{eV} = \frac{1.00e-20}{1.60e-19} = 0.0624 \text{ eV}

Explanation:

This gives the energy uncertainty in more familiar units.

6

Step 6: Calculate Velocity Uncertainty

Assuming an electron, we calculate the velocity uncertainty:

Equation:

Δv=Δpme\Delta v = \frac{\Delta p}{m_e}

Calculation:

Δv=1.00e−259.11e−31=1.10e+5 m/s\Delta v = \frac{1.00e-25}{9.11e-31} = 1.10e+5 \text{ m/s}

Explanation:

This shows how the momentum uncertainty translates to velocity uncertainty.

Frequently Asked Questions (FAQ)

What is the Heisenberg Uncertainty Principle?

The Heisenberg Uncertainty Principle states that there are fundamental limits to the precision with which certain pairs of physical properties (like position and momentum) can be simultaneously known. It's a cornerstone of quantum mechanics.

Why does the uncertainty principle exist?

The uncertainty principle arises from the wave-like nature of matter. When we try to localize a particle, we must use shorter wavelength probes, which impart more momentum to the particle, creating an inherent trade-off between position and momentum precision.

Is the uncertainty principle a limitation of our measuring instruments?

No, the uncertainty principle is not a limitation of our measuring instruments. It's a fundamental property of nature that reflects the wave-particle duality of matter at the quantum level.

What are the practical implications of the uncertainty principle?

The uncertainty principle explains why electrons don't fall into the nucleus, enables quantum tunneling, affects the stability of quantum states, and sets fundamental limits on measurement precision in quantum systems.

How does the uncertainty principle relate to wave-particle duality?

The uncertainty principle is a direct consequence of wave-particle duality. The wave-like nature of particles means they cannot be precisely localized in both position and momentum space simultaneously.

Practice MCQs

  1. The Heisenberg uncertainty principle states:
  2. The uncertainty principle is a consequence of:
  3. Which of the following is NOT an uncertainty relation?
  4. The minimum uncertainty product is:
  5. The uncertainty principle explains why:

Related Calculators

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