At a glance
This sim sends light of one wavelength at a barrier with two narrow slits and records, one arrival at a time, where each lands on a screen. You can compare the real source with classical bullets and change the slit spacing, slit width, wavelength and screen distance. The same rule for adding contributions from different routes explains gratings, X-ray diffraction and the interference of electrons, neutrons and molecules.
What you'll learn
- How the sim combines the two slits for bullets and for the real source, and how those rules differ.
- How the pattern on the screen depends on the slit separation d, the slit width a, the wavelength λ and the screen distance L.
- How to read the detection chart: the bars, the curve and what each is scaled to.
- How a pattern builds up from single detections.
The apparatus from above. A detection lands at some position x along the screen.
What's on screen
- 3D scene: the source, expanding wavefronts, the barrier with its slits labelled S1 (pink) and S2 (violet), and the screen where each detection leaves a dot.
- Blue bars: detections per position bin, scaled so that their total matches the curve's total.
- Green curve: the sim's predicted pattern, scaled so its highest point is 1.
- Orange: the classical bullet curve and bars, shown only when Classical Pattern is on.
See itClear the screen and start a run with both slits open. Watch for: single dots landing one at a time on the screen.
The controls
Press ⌖ on a row to see where that control sits in the sim.
| Control | What it changes | Range | |
|---|---|---|---|
| Slit separation d | Distance between the two slit centres. Moving it clears the screen. | 400–1600 nm opens at 1600 | |
| Slit width a | Width of each slit. With both slits open the sim keeps a below d. Moving it clears the screen. | 100–500 nm opens at 490 | |
| Wavelength λ | Wavelength of the light. Below 380 nm it is shown grey (ultraviolet). It is locked while Classical Pattern is on. | 50–700 nm opens at 525 | |
| Screen distance L | How far the screen stands behind the barrier. Moving it clears the screen. | 4000–8000 nm opens at 4000 | |
| Animation speed | How fast wavefronts expand and detections arrive. The pattern they build does not change. | 5–30 opens at 10 | |
| S1 / S2 | Open or close each slit on its own. The sim does not let you close both. Toggling a slit clears the screen. | both open at the start | |
| Classical Pattern | Switches the screen and chart from the real source to classical bullets, P₁ + P₂, drawn in orange. Switching clears the screen. | top bar opens off | |
| Detection chart | Blue bars of detections against the green curve, or orange against the orange curve. | panel | |
| Play / Pause | Runs the source and detections, or freezes the pattern where it stands. | top bar | |
| Reset | Clears every detection and restarts the run. Your slider settings, slit toggles and Classical Pattern stay as they are. | top bar |
Fixed: the two slits are identical, the light is one wavelength, and the screen covers x = −4000 to +4000 nm.
Bullets add probabilities, the real source adds amplitudes
Both patterns come from the same two slits. They differ in how the two routes are combined. A bullet goes through S1 or S2, so the probabilities add. For the real source each slit contributes an amplitude ψ₁ or ψ₂. The amplitudes add first, and the sum is squared.
The cross term 2 Re(ψ₁*ψ₂) can be positive or negative.
The sim's own curves at the opening settings (d = 1600 nm, a = 490 nm, λ = 525 nm, L = 4000 nm), each scaled to its own peak. Dashed lines mark the five band centres.
- Bullets. P₁ + P₂ is never smaller than either term, so opening the second slit cannot remove a count. At the opening settings the sim draws two separate lobes near x = ±1690 nm and nothing at the centre. Their width comes from the sim's own spray model, not from wave theory.
- Real source. The cross term adds where the two amplitudes arrive in phase and subtracts where they arrive half a cycle apart, so some positions go completely dark. At the opening settings there are five bright bands and the centre is brightest.
See itRun bullets, then the real source, at the opening settings. Watch for: the centre of the screen, empty for bullets and brightest for the real source.
Where the bright bands fall
A bright band needs the two routes to arrive in phase. Their path difference Δ = r₂ − r₁ must then be a whole number of wavelengths. The sim computes Δ exactly from the geometry:
Drawn to scale at the opening settings. The amber stub Δ is the extra distance for the light from S1; at this x it is exactly one λ.
The sim's own path difference, and the conditions for a bright or a dark position.
When L is much larger than d and x, Δ ≈ d sin θ. Neighbouring bands are then about λL/d apart, growing with λ and L and shrinking with d:
Estimates, good only for small angles.
- Accuracy. At the opening settings L is only 2.5 d. The exact |Δ| = λ falls at x = 1414 nm, 8% beyond λL/d = 1313 nm. The brightest spot of the first side band is at 1362 nm, because the envelope in section 05 is falling there.
- Halving d. The first side band moves from 1362 nm to 2794 nm, and the screen goes from five bands to three.
d
a
λ
L
These sliders move the sim too.
See itRun d = 1600 nm, then d = 800 nm. Watch for: the side bands moving outward until only three bands fill the screen.
What sets the height of each band
Each slit has a width a, so it sends light over a range of angles. The sim multiplies the two-slit term, which sets where the bands are, by a single-slit factor, which sets how bright they can be:
Exactly what the sim's green curve is.
Where the envelope vanishes, and when it removes an order.
- Width. A narrower slit gives a wider envelope, so more bands keep their height. At d = 1600, λ = 250, L = 4000 nm the first two side bands reach 0.81 and 0.40 of the centre for a = 400 nm, and 0.87 and 0.57 for a = 320 nm.
- Zeros need a > λ. At the opening settings a/λ = 0.93, so the envelope falls smoothly and never reaches zero. With a ≤ 500 nm, a zero needs λ below 500 nm.
- Missing order. At d = 1500, a = 500, λ = 250 nm (ultraviolet, drawn grey), d/a = 3. Order m = 3, which would sit near x = 2340 nm, falls on the envelope's first zero at 2309 nm and is missing.
See itSet d = 1500 nm, a = 500 nm, λ = 250 nm. Watch for: the green curve lying on zero from about x = 1800 to 3000 nm, with bands visible on either side.
One dot at a time, one amplitude per route
Every detection is a single dot. The pattern appears only as dots pile up, and the bars rise toward the green curve, which is the probability for one dot. In the sim each dot is drawn at random from that probability.
Close S2 and the second amplitude, and the whole cross term, is gone.
From the sim's own curves at the opening settings, each scaled to its own peak. The chart shows one at a time.
- A dark band is not a wall. At x = 678 nm, the first dark position, both slits open give zero. S1 alone gives about 70% of its own peak there. Opening a second route lowers the probability.
- What the sim leaves open. It gives the probability for where dots land. It has no which-slit detector and does not say what happens to one particle between source and screen. That is a question of interpretation.
See itClose S2 at the opening settings. Watch for: the bands giving way to one smooth hump that peaks at x = −800 nm, behind S1.
Where it matters
- Gratings and spectrometers. Many equally spaced slits obey the same condition, d sin θ = mλ, with sharper bands. A grating uses it to separate wavelengths.
- Electrons, neutrons, atoms and molecules. They show the same kind of pattern, with wavelength λ = h/p. A 50 eV electron has λ ≈ 0.17 nm, so the slits must be correspondingly fine.
- Single-particle experiments. Detectors that record one electron at a time see the pattern build from separate dots, as the bars do in the sim.
Go deeper: matter waves, and what the sim assumes
Wavelength from momentum.
Computed from λ = h/p. Eighteen orders of magnitude separate this sim's light from an everyday object.
- A C₆₀ molecule at 200 m/s has λ ≈ 2.8 pm. Interference needs slits comparable to λ, which is why everyday objects show none.
- The sim uses the far-field form of the single-slit factor although L is only a few micrometres. For wide slits and short wavelengths, where a²/λ is comparable to L, a real screen would differ.
- The small-angle formulas in section 04 are estimates. The sim itself uses the exact path difference.
Summary
In one line. The real source adds the two amplitudes before squaring, so the geometry fixes where the bands fall, the slit width sets their heights, and every dot still lands one at a time.
Key points
- Bullets give P₁ + P₂. The real source gives |ψ₁ + ψ₂|², which has a cross term.
- Bright positions satisfy |Δ| = mλ, with Δ = r₂ − r₁. The spacing is about λL/d.
- The single-slit factor sets the heights, with zeros at sin θ = pλ/a when a > λ.
- The pattern is a histogram of single dots drawn from that probability.
Common mix-ups
- "The bands come from particles bumping into each other." The sim draws each dot independently from the probability pattern, and the bands still appear.
- "Narrower slits make narrower bands." The band spacing is set by d, λ and L. The slit width a sets the envelope that scales their heights.
- "The spacing is exactly λL/d." That is a small-angle estimate. At the opening settings the exact first order sits 8% further out.
Things to try on your own
- At a = 490 nm and λ = 525 nm, step d from 1600 nm down to 400 nm and record the first side-band position each time.
- Find a setting where an order goes missing, and check it against d/a = m/p.
- Close S1 instead of S2 and compare the single-slit hump with the one in section 06.