Does the uncertainty principle mean measurement devices are bad?

No. The uncertainty principle is not about imperfect instruments. It is a fundamental property of quantum systems that arises from the wave-like nature of particles and the non-commuting nature of the position and momentum operators. Even with perfect instruments, you cannot make Δx and Δp arbitrarily small at the same time.

When is the equality Δx · Δp = ħ/2 valid?

The equality Δx · Δp = ħ/2 is reached for so-called minimum-uncertainty states, such as Gaussian wave packets. For most realistic states, the product is larger: Δx · Δp > ħ/2.

What units should I use in the uncertainty calculator?

You can choose any consistent set of units. If you enter Δx in meters and mass in kilograms, the calculator will use SI units and return momentum in kg·m/s. If you use nanometers for Δx, the calculator internally converts to meters before applying the formula.

Is the uncertainty principle only about position and momentum?

No. The Heisenberg uncertainty principle is a special case of a more general relation that applies to any pair of non-commuting observables, such as energy and time, or different components of angular momentum. The calculator on this page focuses on the canonical position–momentum pair.

Heisenberg Uncertainty Principle Calculator

Q: What is the Heisenberg uncertainty principle?

The Heisenberg uncertainty principle states that the product of the uncertainties in position and momentum of a particle cannot be smaller than ħ/2. In its most common form: Δx · Δp ≥ ħ/2, where Δx is the standard deviation of position, Δp is the standard deviation of momentum, and ħ is the reduced Planck constant.

Compute the minimum uncertainty in position or momentum for a quantum particle using the Heisenberg relation \( \Delta x \cdot \Delta p \ge \frac{\hbar}{2} \). Includes unit handling, examples, and theory.

Interactive Uncertainty Calculator

What do you want to compute?

Particle mass (optional, for velocity)

Mass is only needed if you want the corresponding velocity uncertainty Δv = Δp / m.

Position uncertainty Δx

Constants used: \( \hbar = 1.054\,571\,817 \times 10^{-34}\,\text{J·s} \), \( c = 2.997\,924\,58 \times 10^{8}\,\text{m/s} \), \( 1\,\text{eV} = 1.602\,176\,634 \times 10^{-19}\,\text{J} \).

Heisenberg Uncertainty Principle: Core Idea

The Heisenberg uncertainty principle is a cornerstone of quantum mechanics. It states that the standard deviations (uncertainties) of position \( \Delta x \) and momentum \( \Delta p \) of a particle obey

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

Here \( \hbar = \frac{h}{2\pi} \) is the reduced Planck constant. The uncertainties are statistical spreads of measurement outcomes, not “errors” of your instruments.

What the calculator does

Mode 1: Given a position uncertainty \( \Delta x \), compute the minimum allowed momentum uncertainty \( \Delta p_{\min} = \frac{\hbar}{2\Delta x} \).
Mode 2: Given a momentum uncertainty \( \Delta p \), compute the minimum allowed position uncertainty \( \Delta x_{\min} = \frac{\hbar}{2\Delta p} \).
Mode 3: Given both \( \Delta x \) and \( \Delta p \), compute the product \( \Delta x \Delta p \) and compare it to \( \hbar/2 \).

If you also provide the particle mass \( m \), the tool converts momentum uncertainty into a velocity uncertainty \( \Delta v = \Delta p / m \).

Formulas Used

1. Position–momentum relation

The canonical Heisenberg relation for position \( x \) and momentum \( p \) is

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

Rearranging gives the minimum uncertainties:

\[ \Delta p_{\min} = \frac{\hbar}{2\,\Delta x}, \qquad \Delta x_{\min} = \frac{\hbar}{2\,\Delta p}. \]

2. From momentum to velocity

For non-relativistic speeds, momentum is \( p = m v \). Therefore, the velocity uncertainty is

\[ \Delta v = \frac{\Delta p}{m}. \]

The calculator uses this relation when you supply a mass. For very high energies or relativistic speeds, this simple relation is only approximate.

3. Units and conversions

Position units: m, nm, pm, fm (internally converted to meters).
Momentum units:
- \(\text{kg·m/s}\) (SI momentum),
- \(\text{eV}/c\): the calculator converts via \( p[\text{kg·m/s}] = \frac{E[\text{eV}] \cdot 1.602\times10^{-19}}{c} \).

Worked Example: Electron in a 1 nm Region

Suppose an electron is localized to about one nanometer:

\( \Delta x = 1\,\text{nm} = 1 \times 10^{-9}\,\text{m} \)
Electron mass \( m_e = 9.11 \times 10^{-31}\,\text{kg} \)

Minimum momentum uncertainty:

\[ \Delta p_{\min} = \frac{\hbar}{2\Delta x} = \frac{1.055\times10^{-34}}{2 \times 10^{-9}} \approx 5.3\times10^{-26}\,\text{kg·m/s}. \]

Corresponding velocity uncertainty:

\[ \Delta v = \frac{\Delta p}{m_e} \approx \frac{5.3\times10^{-26}}{9.11\times10^{-31}} \approx 5.8\times10^{4}\,\text{m/s}. \]

So confining an electron to a nanometer-scale region implies a velocity spread of tens of kilometers per second.

Common Misconceptions

“It’s just about disturbing the particle with measurement”

In early heuristic explanations, the uncertainty principle was sometimes described as a disturbance caused by measurement (e.g., “the photon you use to see the electron kicks it”). Modern quantum theory shows that the principle is deeper: it arises from the non-commuting nature of the operators \( \hat{x} \) and \( \hat{p} \) and the wave-like character of quantum states.

“We could beat it with better instruments”

No matter how perfect your apparatus, the intrinsic spread of outcomes for position and momentum measurements cannot be made arbitrarily small simultaneously. Improving localization (smaller \( \Delta x \)) necessarily increases the spread in momentum outcomes (larger \( \Delta p \)), and vice versa.

“The inequality is always an equality”

The equality \( \Delta x \Delta p = \hbar/2 \) holds only for minimum-uncertainty states (e.g. Gaussian wave packets). For generic states, the product is larger:

\[ \Delta x \Delta p > \frac{\hbar}{2}. \]

Beyond Position and Momentum

The Heisenberg relation is a special case of a more general inequality for any pair of observables \( A \) and \( B \):

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

Examples include:

Energy–time uncertainty, often written \( \Delta E \Delta t \gtrsim \hbar/2 \).
Different components of angular momentum, such as \( L_x \) and \( L_y \).

This calculator focuses on the canonical position–momentum pair, which is the most widely used in introductory quantum mechanics and chemistry.

FAQ

What is the Heisenberg uncertainty principle?

It is a fundamental limit in quantum mechanics that constrains how precisely you can know certain pairs of physical quantities at the same time. For position and momentum, the standard deviations of measurement outcomes obey \( \Delta x \Delta p \ge \hbar/2 \). It is not about bad instruments, but about the structure of quantum theory itself.

Does the uncertainty principle apply to macroscopic objects?

Yes, in principle it applies to all physical systems. However, for macroscopic objects with large mass and relatively large position uncertainties, the implied momentum and velocity uncertainties are so tiny that they are completely negligible in practice. The effects become significant for microscopic systems such as electrons, atoms, and photons.

Why does the calculator talk about “minimum” uncertainties?

The inequality \( \Delta x \Delta p \ge \hbar/2 \) sets a lower bound. For a given \( \Delta x \), the smallest possible \( \Delta p \) is \( \hbar/(2\Delta x) \), but many quantum states have a larger product. The calculator reports this minimum value; real systems may have more uncertainty than the bound requires.

Can I use this for relativistic particles?

The uncertainty relation itself is general, but the simple relation \( p = m v \) used to convert momentum to velocity is non-relativistic. For highly relativistic particles, you should interpret the momentum uncertainty directly, or use a relativistic relation between energy, momentum, and mass.