Quantum Entanglement and Bell Test Simulator: Spooky Action at a Distance

Why does the wave leak? Inside a barrier that is higher than the particle’s energy, the Schrödinger equation has no wavy solution. Instead the wave dies away smoothly, like e^−κx, with κ = √(2m(V₀−E))/ℏ. It never jumps straight to zero, so if the barrier is thin some of the wave is still there on the far side, and from there it travels on as an ordinary wave. The chance of getting through falls off roughly like e^−2κa for a barrier of width a: double the width and the chance is squared.

A ball would bounce back 100%. Classically a particle with less energy than the barrier is always reflected, and a particle with more energy always gets over. Quantum mechanically both rules are broken: waves get through barriers that are too high, and part of a wave reflects even from a barrier it could climb (try “Above the barrier”).

The packet is not split into two particles. |ψ|² gives the probability of where you would find the particle if you looked. After the collision there are two lumps: if you put a detector on the far side, it clicks with probability T and the particle is then entirely there.

Why only certain energies? A wave squeezed between walls must fit: in a box of width L only whole numbers of half-wavelengths fit, so the momentum and therefore the energy come in steps, E_n = n²h²/(8mL²). Smaller box, bigger steps; heavier particle, smaller steps. That is why atoms (tiny boxes for light electrons) have sharp spectral lines, while a marble in a jar seems to have any energy at all.

The shape of the well sets the spacing. Infinite well: gaps grow like n². Harmonic oscillator (a spring): perfectly even steps of ℏω, starting at ½ℏω, never at zero (the zero-point energy). Morse (a real chemical bond): steps that shrink until the molecule falls apart. Finite well: only a few levels fit below the rim, and the waves poke out into the walls a little: that is tunnelling again.

Why does a mix move? Each level’s wave turns in time at its own rate, e^−iE_nt/ℏ. On its own that turning is invisible (|ψ|² stays put: a “stationary state”). Two levels together beat against each other, and the probability sloshes with period h/ΔE. In the double well the two lowest levels are almost equal, so the beat is slow: the particle sits in one well, then tunnels to the other and back, every h/ΔE. The ammonia maser works on exactly this flip-flop of the nitrogen atom.

What you are seeing. The wave goes through both slits; behind the wall the two parts overlap, and where crest meets crest the wave is strong, where crest meets trough it cancels. Each particle is still detected at one single spot. Where it lands is random, but the chance of each spot follows the wave’s brightness, so the stripes appear only after many particles.

Dark stripes are where the paths from the two slits differ by half a wavelength: d sinθ = (m + ½)λ. The whole pattern sits inside the hump of a single slit, whose first zero is at a sinθ = λ.

Pilot-wave paths (the de Broglie–Bohm interpretation) draw one possible path for each particle, steered by the wave with velocity v = (ℏ/m) Im(∇ψ/ψ). Starting points are drawn at random from |ψ|², and the paths bunch into the bright stripes. Standard quantum mechanics predicts the same dots without saying what path the particle took; interpretations differ on that question, not on the measured pattern.

Honest scaling. Real experiments have slits tens of thousands of wavelengths apart, so the stripes are very close together. The wave picture keeps the real shape (the ratio d : a and the number of slits, so the same number of stripes per hump) but draws the wavelength much bigger so you can see it. The screen on the right uses your real numbers. The wall in the picture is a high potential step a few cells thick; real slits are thin plates.

Why it happens. To make a short, sharp bump you have to add up waves of many different wavelengths, and wavelength is momentum (p = h/λ). A long, smooth wave needs only a few. So a packet that is well located (small Δx) has a wide spread of momenta (large Δp). This is a property of waves, not of clumsy measuring.

Let it go. A free packet keeps its momentum spread, so its fast parts run ahead of its slow parts and it spreads out. For any packet whose shape starts out real, Δx(t)² = Δx₀² + (Δp t/m)²; for the Gaussian, Δp = ℏ/(2Δx₀), which gives the formula below. The narrower you make it, the faster it spreads.

Why 2 is the limit for hidden variables. Suppose each pair leaves the source carrying instructions: “if your dial is at a, say +1; at a′, say −1”, and so on, and that Alice’s answer cannot depend on Bob’s dial (locality). Then for each single pair the four products A(a)B(b) − A(a)B(b′) + A(a′)B(b) + A(a′)B(b′) = A(a)[B(b) − B(b′)] + A(a′)[B(b) + B(b′)]. One bracket is 0 and the other is ±2, so every pair contributes exactly +2 or −2, and the average can never be bigger than 2. Bell’s 1964 model (each pair carries a random hidden angle; each side answers + if its dial is within 45° of it for photons, 90° for spins) reaches exactly 2 at the best angles, which is as well as any local model can do there. Quantum mechanics reaches 2√2.

What this simulation is. The quantum source draws each pair’s two results together from the joint probabilities above, with a seeded random number generator. That is the honest way to reproduce quantum predictions on one computer; it is not a model of how nature does it. The two hidden-variable sources really are local: Alice’s answer is computed from her own dial and the shared hidden angle only. Detectors here are perfect and every pair is counted, which real experiments had to work hard to approach (see the loopholes below).

The experiments settle that no theory with local hidden variables and freely chosen settings matches nature. They do not say which assumption to give up, and physicists still disagree:

Non-locality (e.g. de Broglie-Bohm pilot-wave theory, collapse theories): the measurement on one side really does affect the other, but in a way that cannot be used to send messages. Copenhagen-style views: there were no definite values before measurement; the quantum state is a tool for predicting results. Many worlds: every outcome happens, and the correlation shows up when the branches of Alice and Bob meet. Superdeterminism: the settings were never independent of the particles, so the inequality’s assumption fails. Each fits all the data here; the simulator only shows the predictions.

Z then Z: atoms that came out “up” from the first magnet all come out up again. The answer is repeatable.

Z then X: a magnet turned 90° splits them 50/50. Knowing the spin along Z tells you nothing about X.

Z, X, Z: keep only the up-along-X atoms, then measure Z again: half come out down, although every one of them was up along Z two magnets earlier. Measuring X wiped out the Z information.

Recombined: if the X magnet’s two beams are brought back together with nothing recording which one an atom took (a careful “Stern-Gerlach interferometer”, very hard in practice), the original spin comes back intact and the last magnet sends every atom up. Put a detector on the paths (“watched”) and the 50/50 split returns: it is the which-path record, not the magnet, that destroys the information.

What is drawn. The magnets are drawn side on, splitting beams up and down whatever their angle, with the angle shown on each dial; the passing beam is re-centred before the next magnet. The spot sizes and spacing are schematic. The probabilities are exact quantum mechanics; each atom’s path is picked at random from them.

Otto Stern and Walther Gerlach sent a beam of silver atoms from an oven through an uneven magnetic field in Frankfurt in 1922 and collected them on a glass plate. Classical physics expected the atoms’ magnets to point every which way and paint a continuous band. Instead the deposit split into two separate parts. At the time it was read as proof of Bohr and Sommerfeld’s “space quantization”; the real reason, the electron’s spin (a silver atom’s magnetism comes from one unpaired electron), was only proposed by Uhlenbeck and Goudsmit in 1925. Chains of magnets like the ones above are the classic thought experiments used to teach spin (Feynman, Sakurai); with spin-½ atoms the two spots are the whole story.

What you see. A slice shows |ψ|² in one flat plane through the atom: brighter = more likely. The colour is the sign of the wave (orange +, blue −), or with complex m states its phase (the colour wheel turns m times around the z axis). The 3D cloud is a set of random “where was the electron found?” dots, drawn exactly from |ψ|². The radial view adds up all directions: P(r) = r²R² is the chance of finding the electron at distance r.

Nodes. The wave is zero on n − 1 surfaces: n − l − 1 spheres (radial nodes) and l cones or planes (angular nodes). More nodes = more wiggles = higher energy, just like a guitar string.

Bohr’s model and the quantum cloud. Bohr (1913) pictured the electron on circular orbits of radius n²a₀ and got these energies exactly right, which is why the line colours match. But it fails elsewhere: it gives the ground state angular momentum ℏ (the real value is 0: a 1s electron does not circle at all), it cannot explain line brightness or atoms with more electrons, and it says nothing about the shapes above. In the full theory the radius n²a₀ survives only as the most likely radius of the orbitals with l = n − 1 (try 2p, 3d, 4f in the radial view: the peak sits exactly at 4, 9, 16 a₀).