# Heisenberg Uncertainty Principle > The more tightly you pin down where a tiny particle is, the more its speed must spread out, and no instrument can beat that trade-off. Page: https://formulabro.yunusemrevurgun.com/quantum-mechanics/heisenberg-uncertainty-principle/ Category: Quantum Mechanics Published: 2026-10-22 Updated: 2026-10-22 ## Formula $$ \Delta x \, \Delta p \ge \frac{\hbar}{2} $$ Read it out loud: "Delta x times delta p is greater than or equal to h-bar divided by two. The Greek capital letter delta means a spread or a change here. Delta x is the spread in position, which is how far the position measurements scatter. Delta p is the spread in momentum, which is how far the momentum measurements scatter. The symbol with a bar over the h, said h-bar, is the reduced Planck constant. Greater than or equal to means the left side can be equal to the right side or larger, but never smaller." ## Before you start - **Powers of ten, like $10^{-10}$ and $10^{-34}$**: Very large and very small numbers are written as a short number times a power of ten. The small raised number, the **exponent**, says how many times to multiply by 10. A minus sign means divide by 10 instead, so $10^{-10}$ is a decimal point followed by nine zeros and then a 1. An atom is about $10^{-10}$ m across. The number $10^{-34}$ has 33 zeros after the decimal point before the digit 1, which is why Planck's constant looks so tiny. This is called **scientific notation**. - **The Greek capital letter delta, $\Delta$**: Greek letters are extra names for quantities, because the Latin alphabet runs out. The capital delta, written $\Delta$ and said DEL-ta, usually means a change or a spread in physics. So $\Delta x$ means the spread in position measurements, not a number for a single position. Read it as 'delta x', and it is a single symbol with its own meaning, not the letter delta times the letter x. - **Spread, or standard deviation**: When you repeat a measurement many times, the results scatter around an average. The **spread**, also called the standard deviation, measures how far the results typically sit from that average. A small spread means the readings agree closely, and a large spread means they scatter widely. The spread is a property of many measurements taken on identical setups, not an error in one reading. - **Momentum $p$ and the unit kg m/s**: **Momentum** is mass times speed, written $p = mv$. It is the amount of motion an object has, counting how much stuff moves and how fast it moves. A heavy object or a fast object carries a lot of momentum. Momentum is measured in kilogram meters per second, written kg m/s. A 0.145 kg ball moving at 40 m/s has momentum $0.145 \times 40 = 5.8$ kg m/s. - **Planck's constant $h$, the reduced constant $\hbar$, and $\pi$**: Planck's constant, $h = 6.62607015 \times 10^{-34}$ J s, is a fixed number of nature. The symbol $\hbar$, read h-bar, is the same constant divided by $2\pi$, which gives $1.054571817 \times 10^{-34}$ J s. The Greek letter $\pi$ is about 3.14159, the number of diameters that fit around a circle. The number $2\pi$ is about 6.28 and appears whenever something repeats around a circle or a wave completes a cycle. The unit J s means joules multiplied by seconds. - **Wave, wavelength and why a wave has a momentum**: A **wave** is a repeating disturbance that moves through space. Its **wavelength** is the distance from one crest to the next. In quantum physics a particle also has a wave side. A wave squeezed into a small region must be built from many wavelengths, and each wavelength carries its own momentum. Packing a wave into a tighter space therefore spreads out the momentum it carries, which is the heart of this principle. - **The symbol $\ge$, read as greater than or equal to**: The symbol $>$ means greater than, and $=$ means equal. The combined symbol $\ge$ says the left side is either equal to the right side or bigger than it. It never falls below the number on the right. In this formula it says that the product of the two spreads can sit exactly at the limit, or go above it, but cannot go below it. More notation help: https://formulabro.yunusemrevurgun.com/math-basics/ ## What each symbol means - $\Delta x$: The **spread in position**: how widely repeated measurements of the particle's position scatter, measured in meters (m). It is the standard deviation of many position readings on identically prepared particles. For an electron squeezed into an atom, $\Delta x$ is about $10^{-10}$ m, roughly the width of one atom. - $\Delta p$: The **spread in momentum**: how widely repeated momentum measurements scatter, measured in kilogram meters per second (kg m/s). Momentum is mass times speed, so a momentum spread is a spread in how much motion the particle has. In the example, the smallest allowed value of $\Delta p$ is about $5.27 \times 10^{-25}$ kg m/s. - $\ge$: **Greater than or equal to.** The product of the two spreads, $\Delta x \, \Delta p$, can be exactly at the limit or above it, but it can never be smaller than the number on the right. The symbol is read 'is at least'. - $\hbar$: The **reduced Planck constant**, equal to $h / 2\pi = 1.054571817 \times 10^{-34}$ J s, with the unit joule seconds. It is about 6.3 times smaller than $h$. Its tiny size is the reason the trade-off matters only for very light particles in very small spaces. ## The formula piece by piece - $\Delta x$: Start with the position spread, how widely repeated position readings of identical particles are scattered. A tiny spread means the particle is pinned down closely in space, and a large spread means its position is uncertain. - $\Delta p$: Next comes the momentum spread, how widely repeated momentum readings scatter. A tiny spread means the particle's motion is well defined, and a large spread means its motion is uncertain. - $\Delta x \, \Delta p$: Multiply the two spreads to get one number that measures how uncertain position and momentum are together. Squeezing one spread smaller forces the other to grow, so this product cannot be made as small as you like. - $\hbar$: The reduced Planck constant sets the scale of the limit. It is $h$ divided by $2\pi$, and its value of $1.054571817 \times 10^{-34}$ J s is the same for every particle, so it fixes how large the smallest possible product must be. - $\frac{\hbar}{2}$: Half of the reduced Planck constant. This is the smallest value the product of the two spreads can take. Dividing by 2 is part of the standard form of the principle, and the number is about $5.27 \times 10^{-35}$ J s. - $\Delta x \, \Delta p \ge \frac{\hbar}{2}$: The whole formula in one sentence: **the spread in position times the spread in momentum can never be smaller than half the reduced Planck constant.** Squeezing one spread down forces the other up, so the product stays at or above this floor. ## What the formula is doing, and why **What goes in.** Two spreads: $\Delta x$, how widely position measurements scatter, and $\Delta p$, how widely momentum measurements scatter. Both come from repeating a measurement many times on identically prepared particles. **What happens.** The product of the two spreads is multiplied out, and the principle says the result can never be smaller than $\hbar/2$. **What comes out.** A limit. If one spread shrinks, the other must grow so that the product stays large enough. **Why is it a limit of nature and not of instruments?** The principle comes from the wave side of the particle. A wave confined to a small region must be built from a range of wavelengths. Each wavelength carries a momentum $p = h/\lambda$, so the momentum spreads out as the region shrinks. A particle with a sharp position has no sharp momentum, even before anyone measures it. A better ruler or a better detector cannot change that. **Why is the effect so small for most objects?** The floor is $\hbar/2$, and $\hbar$ is extremely small. For a baseball the smallest possible spread in speed is so tiny that it has no practical effect. For an electron confined to the size of an atom, the same floor is large enough to matter, so it sets a real minimum amount of motion. **Why does this matter for atoms?** An electron trapped in an atom is confined to roughly $10^{-10}$ m. The principle then gives a rough lowest momentum and energy for that electron. Pinning the electron into a smaller space costs more energy, which is one reason an atom does not collapse into its nucleus. The example is an estimate that shows the scale of the effect, not a full calculation. **When it applies.** The principle applies to pairs of quantities such as position and momentum, and to a few other pairs in quantum physics. It does not restrict everyday measurements. You can know where a car is to any precision that matters for driving, because its mass is large and its wavelength is far too small to matter. ## An everyday picture Think of a short clap compared with a long tone. A very short clap has no clear pitch, because a clear pitch needs the sound to last long enough to repeat several times. Making the clap shorter blurs its pitch, and a better microphone cannot undo that blur. The same kind of trade-off links position and momentum for a quantum particle: the more tightly its wave is squeezed in space, the more its momentum spreads out. ## Worked example: An electron squeezed into an atom An electron is confined to a region of width $\Delta x = 1.0 \times 10^{-10}$ m, about the width of a small atom. Use $\hbar = 1.054571817 \times 10^{-34}$ J s and the electron's mass $m_e = 9.1093837 \times 10^{-31}$ kg. The principle gives the smallest possible spread in momentum. Dividing that by the mass turns it into the smallest possible spread in speed. 1. **Rearrange the limit to find the smallest momentum spread.** The principle says $\Delta x \, \Delta p \ge \hbar/2$. Dividing both sides by $\Delta x$ gives $\Delta p \ge \frac{\hbar}{2\,\Delta x}$. This tells us the smallest momentum spread allowed for a given position spread, because the product stays at the limit when the two spreads are as small as they can be. 2. **Put in the width of the region.** Twice the width is $2 \times 1.0 \times 10^{-10} = 2.0 \times 10^{-10}$ m. The front numbers give $2.0$, and the power of ten stays at $-10$, because multiplying by 2 does not change any power of ten. 3. **Divide the reduced Planck constant by that number.** The result is $\Delta p \ge \frac{1.054571817 \times 10^{-34}}{2.0 \times 10^{-10}}$. Dividing the front numbers gives $1.054571817 / 2.0 = 0.52729$. For the powers of ten we subtract the exponents, $-34 - (-10) = -24$, so $\Delta p \ge 0.52729 \times 10^{-24}$ kg m/s, which is $5.27 \times 10^{-25}$ kg m/s after moving the decimal point one place. 4. **Convert the momentum spread into a speed spread.** Momentum is mass times speed, so the speed spread is the momentum spread divided by the mass: $\Delta v = \frac{\Delta p}{m_e} \ge \frac{5.27 \times 10^{-25}}{9.1093837 \times 10^{-31}}$. Dividing the front numbers gives about 0.579, and subtracting the exponents, $-25 - (-31) = +6$, gives about $5.79 \times 10^{5}$ m/s. 5. **Compare the speed spread with the speed of light.** Light travels at 299,792,458 m/s, or about $3.0 \times 10^{8}$ m/s. Dividing $5.79 \times 10^{5}$ m/s by that number gives about 0.0019. Multiplying by 100 turns that into 0.19 percent, so the electron's speed must spread out by about one fifth of one percent of the speed of light. 6. **Check the units.** The momentum spread is in kg m/s, and dividing by a mass in kg cancels the kilograms and leaves meters per second. So the answer is a speed spread in the unit we wanted, and the size is consistent with an electron that moves much more slowly than light. **Answer:** The electron's speed must be spread out by at least about $5.79 \times 10^{5}$ m/s, roughly 0.19 percent of the speed of light. Squeezing the electron into atom-sized space means it cannot have one sharp speed. This comes from the wave nature of the particle, not from a poor ruler. ## Common mistakes - Thinking the limit is about bad instruments. A better instrument cannot beat it, because the limit describes the particle's wave, not the measuring device. - Reading $\Delta x$ as the error of one measurement. It is the spread across many measurements on identically prepared particles, not the mistake in a single reading. - Forgetting to divide by the mass when turning a momentum spread into a speed spread. Use $\Delta v = \Delta p / m$, because $p = mv$. - Using $h$ where $\hbar$ belongs. The principle uses $\hbar = h / 2\pi$, which is about 6.3 times smaller than $h$. Using $h$ makes the limit about 6.3 times too large. ## Where you will meet it - Atomic physics, where a quick estimate of the size of an atom and the energy of its electrons comes from this limit. - Squeezed light in gravitational-wave detectors, where the trade-off is used to lower the noise in one measured property at the cost of raising the noise in another. - Wave packets in quantum mechanics, where a particle that is localized in a small region is built from many waves with different momenta. - Textbook problems on confinement, such as an electron trapped in a box a few nanometers wide, where the limit sets a minimum energy. ## Questions people ask ### Is this only a measuring problem? No. Measuring tools do disturb systems, but this principle is a limit on the particle itself. A particle with a sharp position has no sharp momentum, whether or not anyone measures it. ### Does it mean I can never know where something is? You can know where a baseball, a car or a grain of dust is to any precision that matters in daily life. The limit matters only for very light particles confined to very small regions, such as electrons in atoms. ### What does the h-bar symbol mean? $\hbar$, read h-bar, is $h$ divided by $2\pi$. The number $2\pi$ comes from circles and about equals 6.28, so $\hbar$ appears whenever angles or repeating waves are involved. ## Related formulas - [De Broglie Wavelength](https://formulabro.yunusemrevurgun.com/quantum-mechanics/de-broglie-wavelength/) - [Born Rule](https://formulabro.yunusemrevurgun.com/quantum-mechanics/born-rule/) - [Schrödinger Equation](https://formulabro.yunusemrevurgun.com/quantum-mechanics/schrodinger-equation/) ## Sources - [Uncertainty principle, Wikipedia](https://en.wikipedia.org/wiki/Uncertainty_principle) - [Fundamental physical constants, NIST](https://physics.nist.gov/cuu/Constants/) From Formula Bro (https://formulabro.yunusemrevurgun.com/), a side project developed by yunusemrevurgun.com.