A quantum system is prepared in the state |ψ⟩ = (1/√2)|E₁⟩ + (1/√2)|E₂⟩, a superposition of two energy eigenstates with E₁ ≠ E₂. An energy measurement is performed. What happens?
AThe system has both energies E₁ and E₂ simultaneously, and both values are registered
BThe measurement yields the average energy (E₁ + E₂)/2
CThe measurement yields either E₁ or E₂, each with probability 1/2, and the state collapses to the corresponding eigenstate
DThe measurement cannot be performed because the state is not an energy eigenstate
When a measurement is made in quantum mechanics, the outcome is always one of the operator's eigenvalues — never an intermediate value or simultaneous both. The probability of each outcome is |cₙ|², where cₙ = ⟨Eₙ|ψ⟩. Here c₁ = c₂ = 1/√2, so each eigenvalue has probability |1/√2|² = 1/2. After the measurement, the state collapses to the corresponding eigenstate. Option B (the average) is the expectation value ⟨Ĥ⟩ = (E₁ + E₂)/2, which is a statistical property over many measurements — no single measurement ever yields this intermediate value.
Question 2 Multiple Choice
Why must observables in quantum mechanics correspond to Hermitian operators rather than arbitrary linear operators?
AHermitian operators commute with each other, ensuring that two observables can always be measured simultaneously
BHermitian operators are computationally easiest to diagonalize in practice
CHermitian operators guarantee real eigenvalues (so measurement outcomes are real numbers) and orthogonal eigenstates for distinct eigenvalues (so different outcomes correspond to distinguishable states)
DHermitian operators have non-negative eigenvalues, which ensures that probabilities computed from them are non-negative
The two key properties of Hermitian operators ( = †) are: (1) all eigenvalues are real, which is required because physical measurement outcomes are real numbers; (2) eigenstates corresponding to different eigenvalues are orthogonal, which means distinct measurement outcomes correspond to maximally distinguishable quantum states. Option A is wrong — Hermitian operators do not all commute; two Hermitian operators commute if and only if they share a common eigenbasis. Option D is wrong — eigenvalues of Hermitian operators are real but can be negative (e.g., energy eigenvalues can be negative).
Question 3 True / False
A quantum system can be in a state that is not an eigenstate of an observable, but any measurement of that observable will still yield one of the eigenvalues.
TTrue
FFalse
Answer: True
This is the Born rule combined with the spectral theorem. Any state |ψ⟩ can be expanded as a superposition of eigenstates: |ψ⟩ = Σₙ cₙ|φₙ⟩. A measurement always yields one of the eigenvalues λₙ with probability |cₙ|², regardless of whether |ψ⟩ is itself an eigenstate. If |ψ⟩ is not an eigenstate, the measurement outcome is probabilistic — but it is always drawn from the set of eigenvalues, never from an intermediate value. This is why the discreteness of atomic spectra is explained by the energy eigenvalues: only these energies are possible outcomes.
Question 4 True / False
If a quantum state is a superposition of energy eigenstates |E₁⟩ and |E₂⟩, a single energy measurement can yield a value between E₁ and E₂.
TTrue
FFalse
Answer: False
Measurement in quantum mechanics always yields an eigenvalue — never an interpolated or average value. The expectation value ⟨Ĥ⟩ = Σₙ |cₙ|² Eₙ is the statistical average over many measurements, but no individual measurement yields this value unless it happens to equal one of the eigenvalues. This is a fundamental departure from classical physics, where a continuous range of values is typically possible. The discreteness of eigenvalues is what explains discrete atomic spectra: only specific energy transitions (differences between eigenvalues) are possible.
Question 5 Short Answer
What is the physical significance of the completeness of eigenstates of a Hermitian operator, and how does it connect to the probability interpretation of quantum measurement?
Think about your answer, then reveal below.
Model answer: Completeness means the eigenstates of any observable span the full Hilbert space — any quantum state |ψ⟩ can be written as a superposition |ψ⟩ = Σₙ cₙ|φₙ⟩. This is physically significant because it guarantees that a measurement of any observable can always be performed on any state: there is always a well-defined expansion in the eigenbasis. The coefficients cₙ = ⟨φₙ|ψ⟩ are the projections of the state onto each eigenstate, and |cₙ|² gives the probability of obtaining eigenvalue λₙ. Completeness is the mathematical condition that ensures probabilities sum to 1: Σₙ |cₙ|² = ⟨ψ|ψ⟩ = 1.
Without completeness, there could be states that have no expansion in the eigenbasis — states for which the probability interpretation would break down. The spectral theorem for Hermitian operators on Hilbert spaces guarantees completeness, which is why Hermiticity is the physical requirement for observables. Completeness also enables the resolution of the identity Σₙ |φₙ⟩⟨φₙ| = 𝟙, which is the mathematical expression of 'any state can be fully decomposed into measurement outcomes' — the foundation of the Born rule.