# Heisenberg's Uncertainty Principle Calculator

Created by Steven Wooding
Reviewed by Bogna Szyk, Jack Bowater and Adena Benn
Last updated: Oct 27, 2022

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 calculator and velocity calculator 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 (see the De Broglie wavelength calculator) 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. A pretty significant effect on our everyday lives; I hope you agree.

## Heisenberg's uncertainty principle formula

Now that we have some background knowledge, let's take a look at the equation for Heisenberg's uncertainty principle:

$\sigma_x \sigma_p \ge \frac{h}{4\pi},$

where:

• $\sigma_x$ – Standard deviation in the position measurement;
• $\sigma_p$ – Standard deviation in the momentum measurement;
• $h$ – Planck's constant, equal to $\small 6.63 \times 10^{-34}\ \rm J\!\cdot\! s$; and
• $\pi$ – Pi, the ratio of the circumference of a circle to its diameter.

You may also see the formula written as:

$\sigma_x \sigma_p \ge \frac{\hbar}{2},$

where $\hbar$ is the reduced Planck constant, equal to $h/2\pi$.

Up until now, we have been using the term uncertainty. If we measure the properties of a particle, it is more appropriate to talk about the standard deviation (or spread) of values that we obtain from repeated measurements. However, they are very similar in concept, so they are used interchangeably in this article.

Looking at the formula, or inequality, we see that the combined standard deviation of position and momentum has to be greater than Planck's constant divided by four times pi. Notice that Planck's constant is an incredibly small number, meaning that it only really applies when the uncertainties in position and momentum are of a similar magnitude to Planck's constant. We will explore this idea in an example later on in this article.

The Heisenberg uncertainty equation does imply some strange results if you say that you know either the position or momentum with perfect accuracy. Set the uncertainty in momentum to zero, and the uncertainty in position would be infinite. That would mean we are completely unsure of the object's location. It could be anywhere in the universe. And vice versa, we would be infinitely uncertain about the object's momentum (and therefore infinitely uncertain of its mass and speed).

Since these two cases are so extreme, they are not physically possible. You would always have some uncertainty in both the position and the momentum.

## How to use the Heisenberg uncertainty principle calculator

The calculator itself is straightforward to use. Enter a figure for the standard deviation of the position measurement, and it will give you the minimum standard deviation you could hope to obtain of a simultaneous momentum measurement, and vice versa.

If you know the mass of the particle, and it is traveling at non-relativistic speeds (less than half the speed of light), then you can also get an approximate value for the minimum standard deviation of the velocity of the particle. It will technically be a very slight over-approximation since the uncertainty in the momentum is a combination of the uncertainty both in the mass and in the velocity.

In order to avoid getting very small results, please try to keep the uncertainty in position to the order of nanometers, the uncertainty of momentum to around 10‑27 kg·m/s, and mass up to a few hundred atomic mass units (u). You can also use the mass unit me to express the mass in terms of the mass of an electron. See the atom calculator for more details.

## Heisenberg uncertainty principle examples

An electron travels in a straight line at a speed of 2.00 × 106 m/s, well below the speed of light (3.00 × 108 m/s). Your experiment can measure the electron's speed with a precision of 0.5%. Using the uncertainty principle, we can find the minimum precision with which we can measure the electron's position.

First, we calculate the electron's momentum using the equation $p = mv$, where $m$ is the mass of the electron, and $v$ is its velocity or speed. Doing the calculation, we get the result:

$\footnotesize \begin{split} p &= mv\\ &= 9.11\! \times\! 10^{-31}\ \rm kg \times 2.00\! \times\! 10^6\ m/s\\ &= 1.82\! \times\! 10^{-24}\ \rm kg\! \cdot\! m/s \end{split}$

Given that the electron is well below the speed of light, we can assume that the uncertainty in the speed measurement is the same for its momentum. Therefore, we can calculate the standard deviation in the momentum:

$\footnotesize \begin{split} \sigma_p &= 1.82\! \times\! 10^{-24}\ \rm kg\! \cdot\! m/s \times 0.005\\ &= 9.10\! \times\! 10^{-27}\ \rm kg\! \cdot\! m/s \end{split}$

where $\small 0.005$ means 0.5 percent.

Now using the Heisenberg uncertainty principle formula and rearranging for the uncertainty in position $\sigma_x$, we calculate it as:

$\footnotesize \begin{split} \sigma_x &= \frac{6.63\! \times\! 10^{-34}\ \rm J\!\cdot\! s}{4\! \times\! 3.14159\! \times\! 9.10\! \times\! 10^{-27}\ \rm kg\!\cdot\! m/s}\\ &= 5.80\! \times\! 10^{-9}\ \rm m\\ &= 5.8\ \rm nm \end{split}$

What if we try to apply Heisenberg's uncertainty principle to a non-quantum object, such as a baseball? Our baseball is traveling at 92 mph and has been measured with a radar gun with a stated precision of ±1 mph. Since the precision is given as an absolute value, we can simply calculate the uncertainty in the momentum directly, given a baseball weighs 149 grams:

$\footnotesize \begin{split} \sigma_p &= mv = 149\ \rm g \times 1\ mph\\ &= 0.149\ \rm kg \times 1.61\ m/s\\ &= 0.24\ \rm kg\!\cdot\! m/s \end{split}$

Again, we use Heisenberg's uncertainty principle equation to find the minimum uncertainty in position allowed by quantum physics:

$\footnotesize \begin{split} \sigma_x &= \frac{6.63\! \times\! 10^{-34}\ \rm J\!\cdot\! s}{4 \times 3.14159 \times 0.24\ \rm kg\!\cdot\! m/s}\\ &= 2.20\! \times\! 10^{-34}\ \rm m \end{split}$

Wow, that's an incredibly small minimum possible uncertainty in the baseball's position. For comparison, the diameter of a proton is 10‑15 m. This result tells us that for this non-quantum object, we would have no issue with quantum fluctuations when finding the position of the baseball.

Steven Wooding
Position uncertainty (σₓ)
nm
Momentum uncertainty (σₚ)
×10⁻²⁷
kg⋅m/s
Object mass
u
Velocity uncertainty (σᵥ)
m/s
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