Energy-Time Uncertainty Relation

In one sentence: A state that lasts only a short time cannot have one exact energy, so the light it gives off spreads over a small range of colors.

$\Delta E \, \Delta t \gtrsim \frac{\hbar}{2}$

Read it out loud: “Delta E times delta t is greater than or about equal to h-bar divided by two. Delta E is the spread in energy of the state, the capital Greek letter delta followed by E. Delta t is the time scale of the state, such as how long it lasts. The sign that looks like a greater-than sign with a squiggle under it means greater than or about equal to. H-bar is the reduced Planck constant, a fixed tiny number of nature.”

Before you start

The ideas and signs this page uses, explained from zero. For powers of ten, Greek letters and other notation in general, see Math basics.

Delta, $\Delta$ , and spread: The capital Greek letter delta, $\Delta$ , usually means a change or a spread. $\Delta t$ is read as delta t. In this formula $\Delta E$ is the spread in the energy of a state, meaning how wide the range of energies it might have is. A wide spread means the energy is not sharp. Think of the spread as the plus or minus range around the middle value.
Powers of ten, like $10^{-8}$ and $10^{-34}$: Very big and very small numbers are written as a short number times a power of ten. The exponent says how many times to multiply by 10, and a minus sign means divide. So $10^{-8}$ is 0.00000001, with a decimal point, seven zeros and then a 1. Multiplying powers adds the exponents, and dividing subtracts them. So $10^{-27} \div 10^{-19} = 10^{-8}$ .
Joules, electron volts and $\hbar$: Energy is measured in joules, J. An electron volt, eV, is the energy that one electron gains when it is pushed through one volt. One electron volt is $1.602176634 \times 10^{-19}$ J. The reduced Planck constant $\hbar$ , said h-bar, is $1.054571817 \times 10^{-34}$ J s, which means joule seconds, energy times time.
Frequency, hertz and $\nu$: Frequency counts how many waves pass a point each second. The unit is the hertz, written Hz, and 1 Hz means one wave per second. Light at a wavelength of 500 nm has a frequency of about $6.0 \times 10^{14}$ Hz. The Greek letter nu, $\nu$ (said NEW), is the usual symbol for frequency in spectra, and the prefix mega, M, means a million, so 1 MHz is a million hertz.
Excited atoms and the lifetime $\tau$: An atom in an excited state has absorbed energy and sits above its lowest energy level. It falls back, usually within a short time, and sends the extra energy out as light. That light is made of photons, which are single packets of light. The lifetime, written with the Greek letter tau, $\tau$ (said TAU), is the average time the atom stays excited before it falls.
Wavelength $\lambda$ and the speed of light $c$: Wavelength, the Greek letter lambda, $\lambda$ , is the distance from one wave crest to the next. The speed of light, $c$ , is exactly 299,792,458 meters per second. For any wave, speed equals wavelength times frequency, so frequency is $c$ divided by $\lambda$ . A nanometer, nm, is $10^{-9}$ m, so 500 nm is $5.00 \times 10^{-7}$ m.
Greater than or about equal, $\gtrsim$ , and approximately, $\approx$: The sign $\gtrsim$ reads greater than or about equal to. It says that the left side is at least about the right side, with some slack in rough estimates. The sign $\approx$ means approximately equal, which is used when a number has been rounded. Both signs are used because an estimate is rarely exact to the last digit.

What each symbol means

Symbol	Meaning
$\Delta E$	The spread in energy of the state, measured in joules (J), or in electron volts (eV), the unit that is more convenient for atoms. The example gives values near $10^{-27}$ J and $10^{-8}$ eV. A wider spread means the energy is less sharp.
$\Delta t$	A time scale for the state, such as its lifetime, measured in seconds (s). It is the time the state needs to change noticeably or to fall apart. In the example it is $1.0 \times 10^{-8}$ s, which is 10 nanoseconds. A shorter time scale forces a wider energy spread.
$\gtrsim$	The sign reads greater than or about equal to. It has no unit, because it is a comparison and not a quantity. It means the product on the left can be a little smaller than the right side in rough estimates, but the right side sets the scale of the answer.
$\hbar$	The reduced Planck constant, $h/2\pi = 1.054571817 \times 10^{-34}$ J s, in joule seconds. Its unit is energy times time, so $\Delta E\,\Delta t$ also has units of joule seconds. It is a fixed number of nature, and it is tiny because quantum effects are small in daily life.

The formula piece by piece

$\Delta E$: Start with the energy spread, the width of the range of energies. It is the first factor. It is large when the energy is blurred and small when the energy is sharp.
$\Delta t$: The time scale of the state, the second factor. A short lifetime gives a small $\Delta t$ , and a long lifetime gives a large one. The two spreads work against each other in this relation.
$\Delta E\,\Delta t$: Multiply the two spreads. Their product has units of joule seconds, the same units as $\hbar$ . The product cannot drop much below a fixed value, so if one spread gets smaller, the other must grow.
$\frac{\hbar}{2}$: The fixed bound on the product, equal to half of $\hbar$ . Its value is $5.27 \times 10^{-35}$ J s. The one half is a standard result for this bound, so the product is at least about this number.
$\gtrsim$: The sign that says the product is greater than or about equal to the bound. The bound therefore sets the smallest value the product can take, and rough estimates use it as the scale.
$\Delta E \, \Delta t \gtrsim \frac{\hbar}{2}$: The energy spread of a state times its time scale is at least about half of the reduced Planck constant, so a short-lived state must have a wide energy spread.

What the formula is doing, and why

What the relation says. It links two spreads. $\Delta E$ is how far the energy of a state can spread, and $\Delta t$ is a time scale for that state, such as its lifetime. Their product cannot be much smaller than $\frac{\hbar}{2}$ . A short time forces a wide spread in energy, and a long time allows a narrow one. The bound is tiny in everyday units, so the effect matters only for very short-lived or very sharp states.

Why multiply? The relation is a trade-off, not two separate rules. The product has to stay above a fixed number, so making one spread smaller forces the other to grow. The product is the combination that stays fixed, in the same way that a rectangle with a fixed area can be tall and thin or short and wide.

Why time is different from position. In the position and momentum version, both quantities are measured quantities with spreads of values. In standard quantum mechanics, time is not a measured quantity with its own spread. It is the setting that tells the state when to change. So $\Delta t$ is read as a lifetime, or as the time a state takes to change noticeably.

An excited atom. An excited atom stays in its excited state for a short time, then emits a photon and falls to a lower state. Because the state does not last forever, its energy is not one exact number. The light it emits covers a small band of frequencies, and that band is the line width. A spectral line is a narrow band of color in the light from a glowing atom, seen through an instrument that splits light into colors, and its width measures how blurred that color is.

Other widening, and the bound. The bound is a minimum. Real spectral lines are often wider, because collisions between atoms and the motion of the atoms add extra width. Moving atoms emit light at slightly different frequencies, the Doppler effect, which is the same effect that makes a siren sound higher as it comes closer. The lifetime sets the narrowest possible line, called the natural line width. Since $\Delta E = h\,\Delta\nu$ , a lifetime $\tau$ gives a frequency spread of at least about $\frac{1}{4\pi\tau}$ .

An everyday picture

Think of a bell that is struck and then quickly silenced. A short thud has no clear pitch, because a clear pitch needs the sound to last long enough to repeat several times. A bell that rings for a long time has a sharp pitch. The shorter the sound lasts, the wider the range of pitches it contains.

Worked example: The line width of a short-lived excited atom

An excited atomic state has a lifetime of $\tau = 1.0 \times 10^{-8}$ s, which is 10 nanoseconds. This is an assumed round number for the example, not a measured value. In 10 nanoseconds light travels about 3 meters, a little longer than a sofa. Use $\hbar = 1.054571817 \times 10^{-34}$ J s, $h = 6.62607015 \times 10^{-34}$ J s, $1\ \text{eV} = 1.602176634 \times 10^{-19}$ J and $c = 299792458$ m/s.

Write the smallest energy spread. The bound says $\Delta E \gtrsim \frac{\hbar}{2\tau}$ . We use the smallest allowed value, so we set $\Delta E = \frac{\hbar}{2\tau}$ . The lifetime $\tau$ goes in the bottom, multiplied by 2, which gives $2 \times 1.0 \times 10^{-8}$ s.
Compute the energy spread in joules. Divide $\frac{1.054571817 \times 10^{-34}\ \text{J s}}{2 \times 1.0 \times 10^{-8}\ \text{s}}$ . The front numbers give 0.5273, and dividing powers subtracts exponents, $-34 - (-8) = -26$ , so the result is $0.5273 \times 10^{-26}$ J. Tidy it by moving the decimal point one place right and lowering the exponent by one: $5.273 \times 10^{-27}$ J.
Convert to electron volts. One electron volt is $1.602176634 \times 10^{-19}$ J, so divide: $\frac{5.273 \times 10^{-27}}{1.602176634 \times 10^{-19}} \approx 3.29 \times 10^{-8}$ eV. The exponents subtract, $-27 - (-19) = -8$ , and $5.273 / 1.602 \approx 3.29$ . The number is easier to compare with other atomic energies in electron volts.
Convert to a frequency spread. Energy and frequency are linked by $\Delta E = h\,\Delta\nu$ , so $\Delta\nu = \frac{\Delta E}{h} = \frac{5.273 \times 10^{-27}\ \text{J}}{6.62607015 \times 10^{-34}\ \text{J s}}$ . The joules cancel, leaving one per second, which is hertz. The exponents subtract, $-27 - (-34) = +7$ , and $5.273 / 6.626 \approx 0.796$ , so $\Delta\nu \approx 0.796 \times 10^{7} = 7.96 \times 10^{6}$ Hz, about 8 MHz. MHz means megahertz, a million waves per second.
Compare with the light itself. Light at 500 nm has frequency $\nu = \frac{c}{\lambda} = \frac{299792458\ \text{m/s}}{5.00 \times 10^{-7}\ \text{m}} \approx 6.0 \times 10^{14}$ Hz. The spread $7.96 \times 10^{6}$ Hz divided by $6.0 \times 10^{14}$ Hz gives about $1.33 \times 10^{-8}$ . The exponents subtract, $6 - 14 = -8$ . So the spread is about 1.3 parts in 100 million of the light’s own frequency.

Answer: The light from this state covers a frequency range of at least about 8 MHz, or $7.96 \times 10^{6}$ Hz. That is about 1.3 parts in 100 million of the light’s own frequency, so the line is very narrow but not infinitely sharp. A shorter lifetime would give a wider line.

The numbers in this example are recomputed by script every time the site is built.

Common mistakes

Looking for a time measurement with its own spread, as position and momentum have. In standard quantum mechanics time is the setting that drives change, so the relation is read as a lifetime or a time scale.
Treating the bound as the width of every spectral line. Collisions and Doppler motion usually add width on top of the natural line width.
Mixing up joules and electron volts. Convert with $1\ \text{eV} = 1.602176634 \times 10^{-19}$ J before you compare energies.
Dropping the 2 in $\frac{\hbar}{2}$ or the $2\pi$ in $h$ . Either mistake changes the answer by at least a factor of two.

Where you will meet it

Spectroscopy, where the natural width of a spectral line is set by the lifetime of the excited state.
Nuclear and particle physics, where a short-lived particle has a spread in mass set by its lifetime.
Atomic clocks, where long-lived states give very sharp energies and frequencies, which is what makes them good timekeepers.
Quantum physics textbooks, in problems about decaying or unstable states.

Questions people ask

Why is time treated differently from position?

In standard quantum mechanics, position is a quantity with a spread of measured values. Time is the setting that controls how a state changes from moment to moment. That is why the energy-time relation is read as a lifetime or a time scale.

Does this mean an instrument cannot measure the energy of a short-lived state?

A short-lived state has a broad energy, and that breadth is a property of the state itself. Better equipment cannot make the energy sharper than the lifetime allows.

What is a spectral line width?

The width of a spectral line is the range of frequencies, or energies, in the light an atom emits or absorbs. A short lifetime gives a wide line, and a long lifetime gives a narrow one. The width is measured with a spectrometer, an instrument that separates light by color.

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.
Stationary State Time Evolution: A stationary state has one fixed energy, so the chance of finding the particle in each place stays the same over time, even though the wave keeps turning.
Radioactive Decay Law: An unstable sample loses the same fraction of its atoms in each equal stretch of time, so the number left shrinks along a smooth curve.