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Born Rule
The Born rule states that the probability of obtaining a specific measurement outcome is the squared modulus of the corresponding probability amplitude: P(i) = |⟨i|ψ⟩|².
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What it means
The Born rule, proposed by Max Born in 1926, is the fundamental bridge between the mathematical formalism of quantum mechanics and experimental observations.Given a quantum state |ψ⟩ and a measurement in basis {|i⟩}, the probability of obtaining outcome i is P(i) = |⟨i|ψ⟩|².This rule is remarkable because the state vector contains complex amplitudes (with phase information), but measurements yield only real probabilities.The Born rule is considered a postulate of quantum mechanics — it cannot be derived from more basic principles, though there have been many attempts.It applies to all quantum measurements and generalizes to mixed states via the density matrix formalism: P(i) = Tr(ρ|i⟩⟨i|).Everyday analogy
The Born rule is like converting a recipe's ingredient proportions into the probability of tasting each flavor — the proportions (amplitudes) determine the likelihood, but you square them to get the actual probabilities.
Think of it as a lens that converts the complex quantum world into the real-valued probabilities we observe: the quantum state is like an intricate interference pattern, and the Born rule tells you how bright each spot will appear.
Common misconceptions
- The Born rule does NOT say probabilities equal amplitudes — probabilities are the SQUARED MODULUS of the amplitudes, which is critical because amplitudes are complex numbers.
- The Born rule is NOT derived from other quantum postulates — it is itself a fundamental postulate of quantum mechanics, though some interpretations attempt to derive it.
Key takeaways
- The probability of a measurement outcome is the squared modulus of the corresponding amplitude: P(i) = |⟨i|ψ⟩|².
- The Born rule connects the mathematical formalism of quantum states to experimentally observable probabilities.
- Complex phases in amplitudes affect interference patterns but not individual measurement probabilities.
Check your understanding
A qubit is in state |ψ⟩ = (√3/2)|0⟩ + (1/2)|1⟩. According to the Born rule, what is the probability of measuring |1⟩?
- A.1/2
- B.√3/4
- C.1/4
- D.3/4
Show the answer
Answer: C. 1/4
Why: By the Born rule, P(1) = |⟨1|ψ⟩|² = |1/2|² = 1/4.
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Primary source: M. Born, Z. Phys. 37, 863 (1926), doi:10.1007/BF01397477
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