Heisenberg Uncertainty Principle Calculator
Uncertain about the Heisenberg uncertainty principle equation? Then you've come to the right place. You can use this calculator to find the minimum possible uncertainty in either the position or momentum of a quantum-sized object. If you know its mass, you can also find the minimum possible uncertainty in measuring its velocity. This article explains the Heisenberg uncertainty principle definition and why it was so shocking when Heisenberg proposed it.
💡 Our impulse and momentum and velocity calculators might be useful to you.
The Heisenberg uncertainty principle definition
In 1927, Werner Heisenberg proposed a principle that applies to measuring the properties of quantum-sized objects (e.g., atomic and sub-atomic particles). His uncertainty principle states that you cannot measure all of the quantum properties of a particle with the same accuracy at the same time. If your experiment sets out to measure one quantum property with high precision, then you will lose accuracy in the measurement of its other properties.
The two most famous properties that follow Heisenberg's uncertainty principle are position and momentum (mass times speed). The more accurately you set out to measure a particle's location, the more uncertain you are about its momentum (mass and velocity combined) and vice versa.
This uncertainty doesn't come about due to any deficiency in the equipment used to make the measurements. It comes from the concept of wave-particle duality in quantum physics. For example, we all think of an electron as a particle. However, if you fire a load of electrons at two slits in a metal plate, you will get an interference pattern on the other side, as if the electrons were acting like waves. Therefore an electron behaves like a particle and a wave.
You can think of position as a particle-type property and momentum as a wave-type property (as waves are always traveling somewhere). If someone asked you, "what is the position of a wave?" you would have difficulty answering – its position is spread out over a range of values. The situation is similar to measuring momentum. The more accurately you measure the particle's speed, the less sure you can be about its position.
What was shocking about Heisenberg's uncertainty principle was that it made it impossible for experimental science to keep getting more accurate, which was the belief at the time. If the objects were small enough, they would eventually reach a quantum limit of measurement. Another way to look at it is that once you get to the quantum scale, the particles themselves don't know where they are. This uncertainty is sometimes known as quantum fluctuations.
The Heisenberg uncertainty principle can also be applied to other pairs of complementary quantum properties, such as energy and time, and angular position and angular momentum. While the uncertainty principle gets mainly applied to experiments in physics labs, there are some real-world effects. For example, in the Sun's core, it is the uncertainty in the position of two hydrogen nuclei (protons) that allows for there to be a chance that they will overlap and fuse together. This fusing releases a tremendous amount of energy, which we receive as light.
Heisenberg's uncertainty principle formula
The equation for Heisenberg's uncertainty principle is:
σx · σp ≥ h / 4π
- σx — standard deviation (uncertainty) in the position measurement;
- σp — standard deviation (uncertainty) in the momentum measurement;
- h — Planck's constant, equal to 6.62607 × 10−34 J·s; and
- π — Pi, the ratio of the circumference of a circle to its diameter.
You may also see the formula written with the reduced Planck constant ħ (h-bar), where ħ = h / 2π:
σx · σp ≥ ħ / 2
Both forms are identical, since h / 4π = ħ / 2 = 5.2728 × 10−35 J·s. The product of the two uncertainties can never be smaller than this value.
How to use the Heisenberg uncertainty principle calculator
- Choose what you know: the uncertainty in position or the uncertainty in momentum.
- Enter the value and select a unit. Position accepts metric units (m, cm, mm, μm, nm, pm, Å) and American units (in, ft). Momentum accepts kg·m/s, g·cm/s, and lb·ft/s.
- The calculator returns the minimum possible uncertainty in the complementary property.
- Optionally, enter the particle's mass (kg, g, mg, μg, lb, oz) to also get the minimum uncertainty in its velocity (m/s, km/h, ft/s, mph).
Supported unit systems
Metric (SI) units
- Position: meters (m), centimeters, millimeters, micrometers (μm), nanometers (nm), picometers (pm), angstroms (Å)
- Momentum: kilogram-meters per second (kg·m/s), gram-centimeters per second (g·cm/s)
- Mass: kilograms (kg), grams (g), milligrams (mg), micrograms (μg)
- Velocity: meters per second (m/s), kilometers per hour (km/h)
American (imperial) units
- Position: inches (in), feet (ft)
- Momentum: pound-feet per second (lb·ft/s)
- Mass: pounds (lb), ounces (oz)
- Velocity: feet per second (ft/s), miles per hour (mph)
Heisenberg uncertainty principle examples
- Electron in an atom (σx = 0.1 nm): the minimum momentum uncertainty is σp = (5.27 × 10−35) / (1 × 10−10) ≈ 5.27 × 10−25 kg·m/s. With an electron mass of 9.109 × 10−31 kg, the velocity uncertainty is about 5.8 × 105 m/s.
- Dust grain (σx = 1 μm): the minimum momentum uncertainty is ≈ 5.27 × 10−29 kg·m/s — utterly negligible for everyday objects, which is why we never notice the uncertainty principle in daily life.
- Known momentum (σp = 1 × 10−24 kg·m/s): the minimum position uncertainty is σx = (5.27 × 10−35) / (1 × 10−24) ≈ 5.27 × 10−11 m ≈ 0.0527 nm.
Frequently asked questions
Why is there a minimum uncertainty?
The minimum uncertainty is a fundamental property of nature, not a limitation of measuring equipment. It arises from the wave-particle duality of quantum objects. The product σx · σp can never drop below h / 4π.
What is the difference between h and ħ?
h is the Planck constant (6.62607 × 10−34 J·s). ħ ("h-bar") is the reduced Planck constant, equal to h divided by 2π (1.05457 × 10−34 J·s). The uncertainty principle is written as σxσp ≥ h/4π or equivalently σxσp ≥ ħ/2.
Does the uncertainty principle apply to everyday objects?
Technically yes, but the effect is completely negligible at macroscopic scales because Planck's constant is so small. For a baseball or a car, the minimum uncertainty is far smaller than anything that could ever be measured, so classical physics works perfectly.
Can I get the velocity uncertainty without the mass?
No. Momentum is mass times velocity (p = m·v), so to convert a momentum uncertainty into a velocity uncertainty you must divide by the particle's mass. Enter the mass to unlock the velocity result.