A quantum system is in the state |ψ⟩ = (3/5)|a₁⟩ + (4/5)|a₂⟩, where |a₁⟩ and |a₂⟩ are eigenstates of observable  with eigenvalues 2 and 5. What is the probability of measuring the value 5?
A4/5 — the coefficient of the eigenstate corresponding to eigenvalue 5
B16/25 — the square of the coefficient of the eigenstate corresponding to eigenvalue 5
C1/2 — since there are only two eigenstates, each is equally likely
D7/5 — the weighted average of the two eigenvalues
By the Born rule (Postulate 3), the probability of obtaining eigenvalue aₙ is |cₙ|² — the square of the coefficient, not the coefficient itself. The coefficient of |a₂⟩ is 4/5, so the probability is (4/5)² = 16/25. Option (a) is the classic error: confusing the probability amplitude (the coefficient) with the probability (its squared modulus). Option (d) gives the expectation value, which is the average outcome over many measurements, not the probability of any specific outcome.
Question 2 Multiple Choice
Why must quantum mechanical observables be represented specifically by Hermitian operators, rather than general linear operators?
ABecause Hermitian operators are computationally simpler and have well-defined matrix representations
BBecause Hermitian operators always commute with each other, ensuring measurement outcomes are consistent
CBecause Hermitian operators have real eigenvalues, and measurement outcomes must be real numbers
DBecause Hermitian operators preserve the norm of any state vector, ensuring probability is conserved
The requirement of Hermiticity comes directly from physics: when you measure a physical quantity (position, energy, spin), the result must be a real number. Hermitian operators have the mathematical property that all their eigenvalues are real. Non-Hermitian operators can have complex eigenvalues, which cannot represent physical measurement outcomes. Note: norm preservation (option d) is the property of *unitary* operators, which govern time evolution — a separate and equally important requirement.
Question 3 True / False
According to the postulates of quantum mechanics, the time evolution of a quantum state between measurements is fundamentally probabilistic.
TTrue
FFalse
Answer: False
Between measurements, quantum states evolve deterministically and continuously according to the Schrödinger equation (Postulate 4): iℏ d|ψ⟩/dt = Ĥ|ψ⟩. This evolution is unitary and completely predictable given the initial state and Hamiltonian. Probability only enters at the moment of measurement (Postulate 3), where the Born rule governs which eigenvalue is obtained. The deep puzzle of quantum mechanics is precisely this tension: deterministic evolution between measurements, probabilistic discontinuous collapse at measurement.
Question 4 True / False
After a measurement yields eigenvalue aₙ, an immediate second measurement of the same observable on the same system will yield aₙ again with certainty.
TTrue
FFalse
Answer: True
This follows directly from the collapse postulate (Postulate 3). Upon measuring observable  and obtaining eigenvalue aₙ, the state collapses to the corresponding eigenstate |aₙ⟩. Since this state is an eigenstate of  with eigenvalue aₙ, a second measurement immediately after will find the system in |aₙ⟩ with coefficient 1, giving probability |1|² = 1. This repeatability of immediately successive measurements is an experimentally verified consequence of the postulates and distinguishes quantum measurement from classical statistical sampling.
Question 5 Short Answer
What is the conceptual tension between Postulate 3 (measurement and collapse) and Postulate 4 (Schrödinger time evolution), and why is this tension philosophically significant?
Think about your answer, then reveal below.
Model answer: Postulate 4 says the quantum state evolves continuously and deterministically via the Schrödinger equation — given |ψ(t₀)⟩ and Ĥ, the state at any future time is exactly determined. Postulate 3 says measurement causes a discontinuous, probabilistic collapse to an eigenstate — which eigenstate you get is fundamentally random. These two dynamics are inconsistent: a measuring apparatus is itself a physical system subject to Schrödinger evolution, yet the postulates describe it as causing collapse. The theory provides no rule for when 'collapse' occurs versus 'evolution.' This is the quantum measurement problem, and resolving it (or arguing it needs no resolution) is the project of quantum interpretations: Copenhagen, many-worlds, pilot wave, relational, and others.
This tension is not a technical detail — it goes to the heart of what quantum mechanics says about reality. Does the wavefunction represent physical reality or just our knowledge? Does measurement create the outcome or reveal a pre-existing one? The postulates are operationally complete (they predict every experimental result) but ontologically silent on these questions.