Double Slit
V(x, y) = 8(abs(x) < 0.2) · (abs(abs(y) − 1) > 0.4)
Open in the app The dials and keys named below are the app's.
What you see
The formula is the potential over the plane, and it is a product of three things. abs(x) < 0.2 stands a wall 0.4 thick along the y axis. abs(abs(y) − 1) > 0.4 punches two openings in it, each 0.8 wide, centred on y = ±1, so the centres are 2 apart. The 8 is how tall the wall is. Against that the packet's energy is k₀²/2 + 1/2σ² = 4.5 + 0.5 = 5, so the floor under the wall is tinted as ground it has no classical right to enter.
The packet starts at (−4, 0) with Packet width 1 and Momentum k₀ (along x) 3. Its wavelength is 2π/k₀ ≈ 2.09, and it reaches the wall about 1.3 seconds in at Speed 1.
The pattern
Two openings, one wave. Past the wall the pair act as two sources in step, and in directions where the two paths differ by half a wavelength they cancel. With centres d = 2 apart that is sin θ = λ/2d ≈ 0.52, about 32° off the axis, and on an arc of radius 6 the density does fall to near nothing there. It is the only dark line the geometry allows, since the next would need sin θ = λ/d ≈ 1.05. Long waves through close slits give a broad central lobe with one dark line each side, not a train of fringes.
Where T settles
T settles near 0.82, far more than two narrow slits can pass. The wall is thin and not very tall: at an energy of 5 against a height of 8 the wave dies inside it over 1/√(2(8 − 5)) ≈ 0.41, about the wall's own thickness. Much of the packet therefore tunnels straight through it. Seal the openings and T is still 0.55. That leak also fills in the dark lines.
Try
- Change the 8 to 30: the wall goes opaque, T drops to about 0.53, and the dark lines deepen into real nulls.
- Close one slit, 8 · (abs(x) < 0.2) · (abs(y − 1) > 0.4), and the pattern collapses to a single fan.
- Set Momentum k₀ (along x) to 2: the wavelength grows to π ≈ 3.14 and the dark line swings out past 50°.
- Press Top for the plan view, where the two dark lines are easiest to place.