Double-Slit Interference

In one sentence: Waves from two narrow openings overlap, and at certain places they add into bright bands while at others they cancel into dark ones.

$d \sin\theta = m\lambda$

Read it out loud: “d times sine theta equals m times lambda. The letter d is the distance between the centers of the two slits. Sine theta is the sine of the angle theta, which is the angle from the center line to a bright band. The letter m is a whole number, 0, 1, 2 and so on, giving the order of the band. The Greek letter lambda is the wavelength, the distance from one wave crest to the next.”

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.

Waves, wavelength $\lambda$ and crests: A wave is a repeating disturbance that carries energy from place to place without carrying its material along with it. A crest is its highest point and a trough is its lowest point. The wavelength, written with the Greek letter lambda, $\lambda$ (said LAM-duh), is the distance from one crest to the next. Light, water ripples and sound are all waves.
Interference: Interference happens when two waves overlap. Where two crests arrive together, they add and make a bigger wave, which we see as a bright spot. Where a crest meets a trough, they cancel and make a dark spot. Two waves that arrive with a gap of a whole number of wavelengths add up, and a gap of half a wavelength makes them cancel.
Sine of an angle, degrees and radians: The sine of an angle, written $\sin\theta$ , is a ratio from a right triangle: the side opposite the angle divided by the longest side. For small angles the sine is almost the same as the angle in radians, so $\sin 1^\circ \approx 0.0175$ . A full turn is 360 degrees, or $2\pi$ radians, and $\pi$ radians is 180 degrees.
Powers of ten, like $10^{-3}$ and $10^{-7}$: A power of ten shows how many times to divide by 10. The power $10^{-3}$ is 0.001, and $10^{-7}$ is 0.0000001, which is a decimal point followed by six zeros and then a 1. Multiplying powers adds the exponents, and dividing subtracts them. So $5.00 \times 10^{-7} \div 5.0 \times 10^{-4} = 1.0 \times 10^{-3}$ .
Units: nanometer, millimeter and meter: The meter, m, is the base unit of length. A millimeter, mm, is one thousandth of a meter, which is $10^{-3}$ m. A nanometer, nm, is one billionth of a meter, which is $10^{-9}$ m. Light in the green part of the rainbow has a wavelength of about 500 nm, which is $5.00 \times 10^{-7}$ m.
Whole numbers and the order $m$: A whole number is 0, 1, 2, 3 and so on, and it can be negative, like $-1$ or $-2$ . The path difference is the extra distance one wave travels compared with the other, and it is measured in whole wavelengths for a bright band. The order $m$ counts those wavelengths. The first bright band has $m = 1$ , the second has $m = 2$ , the center band has $m = 0$ , and bands on the other side have negative $m$ .
Delta $\Delta$ and the ratio $y/L$: The capital Greek letter delta, $\Delta$ , means a change, so $\Delta y$ is the distance between two neighboring bright bands on the screen. A ratio such as $\frac{y}{L}$ is one number divided by another. Here $y$ is the sideways distance on the screen from the center, and $L$ is the distance from the slits to the screen. For a small angle, dividing them gives a small number.
Wave function $\psi$ and probability: The wave function $\psi$ , said psi, describes a particle as a wave, and it is a complex number at each place. Its squared size, $|\psi|^2$ , gives the probability of finding the particle at that place. A probability is a chance written as a number from 0 to 1, where 0 means never and 1 means always. The bright bands are the places where the probability is high.

What each symbol means

Symbol	Meaning
$d$	The distance between the centers of the two slits, in meters (m). It is not the width of one slit. In the example it is 0.50 mm, which is $5.0 \times 10^{-4}$ m. A smaller $d$ spreads the pattern wider on the screen.
$\theta$	The angle from the center line to a bright band, measured from the slits. Written as a ratio it has no unit, but as an angle it is in radians (rad) or degrees. For a small angle, the value in radians is almost the same as the sine. In the example the first bright band sits at about $1.0 \times 10^{-3}$ rad, which is about 0.057 degrees.
$m$	The order of the bright band: a whole number 0, 1, 2 and so on, with a sign for the side of the center. It has no unit, because it counts wavelengths. It is not a mass, even though it uses the letter m, and the mass of the particle does not appear in this formula.
$\lambda$	The wavelength of the wave, the distance from one crest to the next, in meters (m). In the example it is 500 nm, which is $5.00 \times 10^{-7}$ m, in the green part of visible light. A longer wavelength spreads the bands farther apart.
$\Delta y$	The distance between neighboring bright bands on the screen, in meters (m). It is the spacing that the example finds, $1.0 \times 10^{-3}$ m, which is 1.0 mm. It is used for small angles, where the bands are evenly spaced.
$L$	The distance from the slits to the screen, in meters (m). In the example it is 1.0 m. A longer distance spreads the bands farther apart, because the spacing grows in step with $L$ .

The formula piece by piece

$d$: Start with the slit spacing, the distance between the centers of the two openings. It sets how far apart the two waves start, and so how big the path difference can get at a given angle.
$\sin\theta$: The sine of the angle to the band. It is the ratio that links the geometry of the screen to the spacing of the slits, so it turns the slit spacing into the extra distance.
$d\sin\theta$: The path difference, which is the extra distance one wave travels compared with the other. Multiplying the slit spacing by the sine gives this extra distance in meters.
$m\lambda$: A whole number of wavelengths. When the path difference equals $m$ wavelengths, the two waves arrive in step and add to make a bright band.
$d \sin\theta = m\lambda$: A bright band appears at an angle where the extra distance from one slit compared with the other equals a whole number of wavelengths.
$\Delta y = \frac{\lambda L}{d}$: For small angles the bright bands are evenly spaced on the screen. The spacing equals the wavelength times the screen distance, divided by the slit spacing.

What the formula is doing, and why

Two waves, one pattern. A wave passing through two narrow slits spreads out from each slit. The two spreading waves overlap on a screen. Where a crest meets a crest, the waves add and the spot is bright. Where a crest meets a trough, they cancel and the spot is dark. The bright spots are called fringes, or bands.

Why sine? The rule for a bright band is $d\sin\theta = m\lambda$ . Here $d\sin\theta$ is the extra distance one wave travels compared with the other. You can see where the sine comes from with a right triangle whose longest side is the slit spacing $d$ and whose angle is $\theta$ . The side opposite that angle has length $d\sin\theta$ , and that side is the extra distance. When it equals a whole number of wavelengths, the two waves arrive in step and add.

Why dark bands sit between. Dark bands happen when the extra distance is halfway between whole numbers of wavelengths: $d\sin\theta = \left(m + \frac{1}{2}\right)\lambda$ . Then the crest of one wave meets the trough of the other, and they cancel. Half a wavelength is exactly the amount needed for that cancellation.

Why the spacing is even. For small angles, $\sin\theta \approx \tan\theta = y/L$ , so the bright bands are evenly spaced: $\Delta y = \frac{\lambda L}{d}$ . Why multiply by $\lambda$ and $L$ ? A longer wavelength or a longer distance to the screen spreads the pattern out. Why divide by $d$ ? Slits placed farther apart pack the pattern closer together, because each band sits at a smaller angle.

Particles too. The same pattern appears when particles pass through the slits one at a time. Each particle lands at a single spot, but the spots build up over time into the fringe pattern. The wave function predicts the chance of landing at each place, and that chance follows the bright and dark bands. Thomas Young’s interference experiments with light in the early 1800s helped show that light behaves as a wave. The same pattern for electrons is strong evidence that matter also has wave behavior.

An everyday picture

Think of two stones dropped into a still pond at the same time. Ripples spread out from each stone and overlap. In some spots the peaks meet and make a tall wave, and in others a peak meets a dip and the water stays flat. The double-slit pattern is the same overlap of ripples, drawn on a screen.

Worked example: Fringe spacing for green light

Green light of wavelength $\lambda = 500$ nm passes through two slits with $d = 0.50$ mm. The screen is $L = 1.0$ m away. Find the spacing between bright bands on the screen, and the angle of the first bright band, which has order $m = 1$ . The values are chosen for this example.

Convert every length to meters. The formula needs all lengths in the same unit. A nanometer is $10^{-9}$ m, so $\lambda = 500 \times 10^{-9}$ m $= 5.00 \times 10^{-7}$ m. A millimeter is $10^{-3}$ m, so $d = 0.50 \times 10^{-3}$ m $= 5.0 \times 10^{-4}$ m. The screen distance is already $L = 1.0$ m.
Find the spacing between bright bands. Use $\Delta y = \frac{\lambda L}{d}$ . Multiply the top: $5.00 \times 10^{-7}$ m $\times 1.0$ m, which is meters times meters. Divide by the bottom, $5.0 \times 10^{-4}$ m, and the meters cancel to leave meters. The number is $\frac{5.00 \times 10^{-7}}{5.0 \times 10^{-4}} = 1.0 \times 10^{-3}$ m, because the exponents subtract, $-7 - (-4) = -3$ , and $5.00 / 5.0 = 1.0$ .
Convert the spacing to millimeters. $1.0 \times 10^{-3}$ m is 1.0 mm, because a millimeter is $10^{-3}$ m. So the bright bands are 1.0 mm apart on the screen, a spacing that a ruler can measure.
Find the sine of the first bright band. The first bright band has $m = 1$ , so $d\sin\theta = \lambda$ . Dividing both sides by $d$ gives $\sin\theta = \frac{\lambda}{d} = \frac{5.00 \times 10^{-7}}{5.0 \times 10^{-4}} = 1.0 \times 10^{-3}$ . The exponents subtract the same way, $-7 - (-4) = -3$ .
Find the angle. For such a tiny sine, the angle in radians is almost the same number, so $\theta \approx 1.0 \times 10^{-3}$ rad. To convert to degrees, multiply by $\frac{180}{\pi} \approx 57.3$ , which gives $1.0 \times 10^{-3} \times 57.3 \approx 0.0573$ degrees.
Check the small-angle idea. The angle is far below one degree, so the rule that the bands are evenly spaced is accurate here. The first bright band sits only about 0.06 degrees from the center line.

Answer: On a screen 1 m away, the bright bands are about 1.0 mm apart. The first bright band sits at an angle of about 0.057 degrees from the center. The small angle is why the even spacing formula works here.

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

Common mistakes

Mixing units, such as using 500 for nanometers without converting to meters. Every length must be in meters before the formula is used.
Reading $d$ as the width of each slit. It is the distance between the slit centers.
Using $\Delta y = \frac{\lambda L}{d}$ for large angles. That spacing formula needs the angle to be small, so for larger angles use the exact rule $d\sin\theta = m\lambda$ .
Confusing bright and dark bands. Bright bands need a whole number of wavelengths, and dark bands need a whole number plus one half, such as $\frac{1}{2}$ or $\frac{3}{2}$ .

Where you will meet it

Interference demonstrations in physics teaching labs, following Young’s experiment.
Diffraction gratings in spectrometers, which use the same bright-band rule with many slits.
Electron and neutron interferometry, which confirm the wave behavior of matter.
Optical measurements of very small distances, where the fringe spacing reveals a wavelength or a slit separation.

Questions people ask

If particles arrive one at a time, why does a pattern form?

Each particle lands at one spot, and the chance of landing at each spot follows the wave pattern. Many single arrivals build up the bright and dark bands.

Does the pattern come from particles hitting each other?

No. The pattern still forms when particles pass through one at a time, so it is not caused by collisions between them.

What happens if one slit is covered?

The bands disappear, and the screen shows a single broad spread instead. Both paths must be open for the interference bands to appear.

Is $m$ the same as mass?

No. In this formula $m$ is the order of the band, which is a whole number such as 0, 1 or 2. The mass of the particle does not appear in the formula.

De Broglie Wavelength: Every moving object has a wave that goes with it, and the heavier or faster the object moves, the shorter that wave is.
Born Rule: The Born rule turns a quantum wave into chances: the probability of a result is the squared size of the wave for that result, so bigger waves mean more likely results.
Bragg's Law: X-rays bounce off the evenly spaced layers of a crystal, and the bounces add into one strong peak only at certain angles.