Double-Slit Interference

The angles of the maxima follow from d·sin(α_k) = k·λ: sin(α_k) = k·λ/d with order k = 0, 1, 2, ... The zeroth maximum always lies straight ahead in the centre. Example: d = 0.1 mm and λ = 600 nm give for k = 1 the value sin α = 6×10⁻⁷/10⁻⁴ = 0.006, so α ≈ 0.34°. The position on a screen at distance L follows from geometry: x = L·tan α, for small angles x ≈ k·λ·L/d, here 12 mm. The maxima are then equally spaced. Convert all lengths consistently to metres; mixing nm, µm and mm is the most common source of error.

The path difference Δs is the difference in distance the waves travel from the two slits to a point on the screen; for distant points Δs = d·sin α. It determines the relative timing of the two waves there. If Δs is a whole multiple of the wavelength (0, λ, 2λ, ...), crest meets crest: constructive interference, a bright maximum. If Δs is a half-integer multiple (λ/2, 3λ/2, ...), crest meets trough and the waves cancel: a dark minimum. Everything in between produces intermediate brightness. That light plus light yields darkness at the dark places is the decisive wave argument; particle trajectories alone could never explain it.

You measure lengths and convert back to nanometres; that is the core of the Young experiment. Illuminate the double slit with the light to be measured, ideally a laser, and measure three quantities: the slit separation d, the screen distance L and the distance x of the first maximum from the centre. For small angles λ = d·x/(k·L). Example: d = 0.2 mm, L = 2 m, first maximum at x = 6.4 mm: λ = 2×10⁻⁴ × 6.4×10⁻³/2 = 6.4×10⁻⁷ m = 640 nm, red light. It becomes more accurate if you average over several orders or measure the distance across many maxima and divide.

From sin α = k·λ/d both trends can be read off directly. A larger wavelength spreads the pattern: red light (about 650 nm) produces maxima farther apart than blue (about 450 nm); with white light this creates coloured fringes, blue on the inside, red on the outside. A smaller slit separation also spreads the pattern, so a finer double slit separates better. Conversely, for large d the maxima crowd together until they merge, which is why you never observe interference from two windows. In addition sin α ≤ 1 limits the number of orders to k ≤ d/λ: for d = 4 µm and λ = 500 nm only orders up to k = 8 exist.

Yes, and it is one of the most important experiments of quantum physics. Sending electrons through a double slit produces the same fringe pattern on the detector as light, described by the same formula d·sin α = k·λ, only with the de Broglie wavelength λ = h/p of the electron. The astonishing part: the pattern builds up even when the electrons fly one at a time; each electron in a sense interferes with itself. If, however, you measure which slit the electron passes, the pattern disappears. The experiment thus demonstrates wave-particle duality and the role of measurement, and in final exams it is the standard bridge from optics to quantum physics.