Quantum Superposition

Quantum superposition

A quantum state |Ψ⟩ can be a linear combination of basis states:

|Ψ⟩ = α|0⟩ + β|1⟩

α and β are complex amplitudes, satisfying |α|² + |β|² = 1 (normalization).

Born rule: probability of finding the system in state |0⟩ on measurement is |α|²; in |1⟩ is |β|².

Common superposition states

State	Form	Notes
|0⟩	α=1, β=0	"Pure 0" basis state
|1⟩	α=0, β=1	"Pure 1"
|+⟩ = (|0⟩+|1⟩)/√2	α=β=1/√2	Equal superposition
|−⟩ = (|0⟩−|1⟩)/√2	α=−β=1/√2	Equal but anti-phase
(|0⟩+i·|1⟩)/√2	α=1/√2, β=i/√2	Complex amplitude

Interference

If a particle has two paths to reach a detector, with amplitudes A₁ and A₂, total amplitude is A_total = A₁ + A₂. Probability is |A_total|².

Phase relationship	P_total
In phase (same)	|A₁ + A₂|² = (A + A)² = 4A² (constructive)
Out of phase (opposite)	|A − A|² = 0 (destructive)
Random	|A|² + |A|² = 2A² (no interference)

JavaScript — superposition

// Complex number basics
class Complex {
  constructor(re, im = 0) { this.re = re; this.im = im; }
  add(c) { return new Complex(this.re + c.re, this.im + c.im); }
  mul(c) { return new Complex(this.re*c.re - this.im*c.im, this.re*c.im + this.im*c.re); }
  abs2() { return this.re*this.re + this.im*this.im; }
}

// Two-state superposition
function superposition(alpha, beta) {
  // alpha, beta as Complex; |α|² + |β|² should = 1
  const norm = alpha.abs2() + beta.abs2();
  return { alpha, beta, norm };
}

// Equal superposition (Hadamard-like)
const equal = superposition(new Complex(1/Math.sqrt(2)), new Complex(1/Math.sqrt(2)));
console.log(`P(0) = ${equal.alpha.abs2()}`);  // 0.5
console.log(`P(1) = ${equal.beta.abs2()}`);   // 0.5

// Quantum measurement collapse
function measure(state) {
  if (Math.random() < state.alpha.abs2()) return 0;
  return 1;
}

// Statistical sample
const counts = [0, 0];
for (let i = 0; i < 10000; i++) {
  counts[measure(equal)]++;
}
console.log(`Outcomes: 0 = ${counts[0]}, 1 = ${counts[1]}`);
// ~5000, 5000 — confirms |α|² and |β|² probabilities

// Interference: combining two paths
function combineAmplitudes(amplitudes) {
  // Sum complex amplitudes coherently
  return amplitudes.reduce((sum, a) => sum.add(a), new Complex(0, 0));
}

// Two paths in phase
const path1 = new Complex(1/Math.sqrt(2));
const path2 = new Complex(1/Math.sqrt(2));
const total_inphase = combineAmplitudes([path1, path2]);
console.log(`In-phase: |A|² = ${total_inphase.abs2()}`);  // 2 (constructive)

// Opposite phase
const path2_neg = new Complex(-1/Math.sqrt(2));
const total_outphase = combineAmplitudes([path1, path2_neg]);
console.log(`Out-of-phase: |A|² = ${total_outphase.abs2()}`);  // 0 (destructive)

// Random phase (no interference)
const path2_phaseRandom = new Complex(0, 1/Math.sqrt(2));  // 90° offset
const total_random = combineAmplitudes([path1, path2_phaseRandom]);
console.log(`90° phase: |A|² = ${total_random.abs2()}`);  // 1 (no interference)

// Hadamard gate (creates equal superposition from |0⟩)
function hadamard(state_alpha, state_beta) {
  // H = 1/√2 · [[1, 1], [1, -1]]
  const a = (state_alpha + state_beta) / Math.sqrt(2);
  const b = (state_alpha - state_beta) / Math.sqrt(2);
  return [a, b];
}

console.log(`H|0⟩ = ${hadamard(1, 0)}`); // [0.707, 0.707]
console.log(`H|1⟩ = ${hadamard(0, 1)}`); // [0.707, -0.707]
console.log(`H·H|0⟩ = ${hadamard(...hadamard(1, 0))}`); // [1, 0] — back to |0⟩

Where superposition matters

• Quantum computing. Qubits in superposition allow exponential parallelism for certain problems.
• Quantum cryptography. BB84 protocol uses superposition for secure key distribution.
• Atomic clocks. Atoms placed in superposition states; precise frequencies measured via Ramsey interferometry.
• NMR/MRI. Nuclear spins put in superposition; magnetic resonance signals from coherent precession.
• Quantum sensing. Superposition enhances sensitivity (Heisenberg-limited measurements).
• Foundations of QM. Bell tests, GHZ states test classical vs quantum predictions.
• Materials. Superconductors — Cooper pairs in coherent superposition; Bose-Einstein condensates.

Common mistakes

• Thinking superposition means classical "or." Classical: "system is in A OR B, I just don't know which." Quantum: "system is in α·A + β·B — both contribute simultaneously, give interference effects."
• Believing measurement is necessary for collapse. Decoherence (interaction with environment) effectively collapses superpositions even without explicit measurement. Lab quantum systems must be isolated.
• Treating amplitudes as probabilities. Amplitudes can be negative or complex; probabilities are |amplitude|². Negative amplitudes enable interference.
• Forgetting normalization. |α|² + |β|² = 1 always. If you change one, the other must adjust.
• Treating superposition as classical mixture. Mixed state and pure superposition give different outcomes statistically.
• Confusing entanglement with superposition. Superposition — single particle in multiple states. Entanglement — multiple particles in correlated states.