A particle is in state |ψ⟩ = (3/5)|a₁⟩ + (4/5)|a₂⟩. What is the probability of measuring eigenvalue a₁?
A3/5, because that is the coefficient of the eigenstate |a₁⟩
B9/25, because the Born rule gives the modulus squared of the amplitude
C1/2, because there are only two possible outcomes and they must be equally likely
D4/5, because the larger coefficient dominates the measurement outcome
The Born rule states P(aₙ) = |⟨aₙ|ψ⟩|². The amplitude for a₁ is 3/5, so the probability is |3/5|² = 9/25. The amplitude itself (3/5) is not the probability — this is the most common error. Note that P(a₂) = |4/5|² = 16/25, and 9/25 + 16/25 = 1, confirming completeness. The amplitude is a complex number that carries phase information; the probability discards the phase by taking the modulus squared.
Question 2 Multiple Choice
After measuring a particle's spin and finding it to be spin-up, a student argues that the Schrödinger equation will now evolve the state back into a superposition of spin-up and spin-down over time. What is correct about this reasoning?
AThe student is correct — the Schrödinger equation always eventually restores a superposition
BThe Schrödinger equation governs unitary evolution between measurements; state collapse upon measurement is a separate postulate that instantly projects the state into the measured eigenstate
CThe student is correct only if the spin-up state is not an energy eigenstate
DThe student is wrong because superpositions only arise for particles with many degrees of freedom
This gets at a deep feature of quantum measurement. The Schrödinger equation describes smooth, continuous, unitary evolution of the wavefunction between measurements — it never produces collapse. State collapse is a separate postulate: upon measurement yielding aₙ, the state instantly becomes |aₙ⟩. This discontinuity is not derived from the Schrödinger equation; it is an additional rule. The post-measurement state |↑⟩ will then evolve unitarily under Schrödinger — but that evolution will only produce a superposition again if the Hamiltonian mixes spin states (e.g., in a magnetic field in a different direction).
Question 3 True / False
If a system is already in an eigenstate of the observable being measured, the Born rule predicts that measurement yields the corresponding eigenvalue with probability 1.
TTrue
FFalse
Answer: True
If |ψ⟩ = |aₙ⟩, then ⟨aₙ|ψ⟩ = ⟨aₙ|aₙ⟩ = 1, and for all m ≠ n, ⟨aₘ|ψ⟩ = ⟨aₘ|aₙ⟩ = 0 by orthogonality. So P(aₙ) = 1 and P(aₘ) = 0 for all other eigenvalues. This is consistent with the collapse postulate: if you measure a system already in an eigenstate, you get that eigenvalue with certainty, and the state after measurement is unchanged.
Question 4 True / False
The probability amplitude ⟨aₙ|ψ⟩ directly gives the probability of measuring eigenvalue aₙ.
TTrue
FFalse
Answer: False
The probability amplitude ⟨aₙ|ψ⟩ is a complex number, not a probability. Probabilities must be real and between 0 and 1. The Born rule takes the modulus squared: P(aₙ) = |⟨aₙ|ψ⟩|². Taking the modulus squared discards phase information and produces a real non-negative number. The amplitude itself carries phase, which is physically meaningful for interference phenomena — but it is not directly observable. Confusing amplitude with probability is one of the most common errors in introductory quantum mechanics.
Question 5 Short Answer
Why is state collapse described as a separate postulate from the Schrödinger equation, and what experimental consequence demonstrates that collapse actually occurs?
Think about your answer, then reveal below.
Model answer: The Schrödinger equation describes unitary, continuous, deterministic evolution — it never causes a state to jump discontinuously into a single eigenstate. Collapse is a separate, non-unitary postulate. The experimental consequence: if you immediately repeat the same measurement after obtaining eigenvalue aₙ, you always get aₙ again. This is consistent with the collapsed state being |aₙ⟩, which yields aₙ with probability 1.
If there were no collapse — if the state continued to evolve unitarily after measurement — then a second identical measurement would generally not give the same result, because the state would have evolved away from the eigenstate. The fact that repeated measurements give the same result (immediately after the first) is direct experimental evidence for collapse. This is also why measuring in a different basis after the first measurement generally gives random results: the collapse into |aₙ⟩ creates a definite state in the original basis, but that state is typically a superposition in other bases.