A particle is in the state ψ = (1/√2)(ψ₁ + ψ₂), where ψ₁ and ψ₂ are energy eigenstates with eigenvalues E₁ and E₂. What result will a single measurement of energy yield?
AThe average energy (E₁ + E₂)/2, since the particle is equally in both states
BEither E₁ or E₂, with equal probability 1/2 each, and the state collapses to the corresponding eigenstate
CBoth E₁ and E₂ simultaneously, since the particle is in a superposition of both
DAn undefined result, because the energy operator cannot act on a superposition state
A superposition state does not have a definite energy — this is the foundational departure from classical mechanics. A single measurement yields exactly one eigenvalue (either E₁ or E₂) with probabilities determined by the squared amplitudes of the coefficients. With equal coefficients 1/√2, each outcome has probability (1/√2)² = 1/2. After measurement, the state collapses to the corresponding eigenstate. Option A describes the expectation value ⟨E⟩ = (E₁ + E₂)/2 — the average over many measurements — which is not the result of any single measurement.
Question 2 Multiple Choice
Which property of Hermitian operators is essential for ensuring that quantum mechanical observables yield physically meaningful measurement outcomes?
AHermitian operators have a finite number of eigenvalues, making the set of possible outcomes discrete and countable
BThe eigenvalues of Hermitian operators are always real numbers, consistent with measurement results being real
CHermitian operators commute with each other, allowing simultaneous measurement of all observables
DHermitian operators always have normalized eigenstates, making probability calculations straightforward
Physical measurements always yield real numbers. A Hermitian operator  satisfies † = Â, which guarantees all its eigenvalues are real. This is why physical observables — position, momentum, energy — must be represented by Hermitian operators. Option C is incorrect: Hermitian operators do *not* in general commute (the canonical non-commutation is [x̂, p̂] = iℏ), and non-commuting observables cannot be simultaneously measured with certainty. Option A is false: many Hermitian operators (like position) have continuous, not discrete, spectra.
Question 3 True / False
The momentum operator p̂ = −iℏ ∂/∂x returns a definite momentum value when applied to any normalizable wavefunction.
TTrue
FFalse
Answer: False
The eigenvalue equation p̂ψ = pψ holds only for eigenstates of p̂ — functions of the form e^{ikx} with definite momentum ℏk. For a general superposition wavefunction, applying p̂ does not return a scalar multiple of the same function; it returns a different function. What can be extracted for a general state is a probability distribution over momentum eigenvalues, not a single definite value. The common misconception is that applying an operator to any state yields an eigenvalue; in fact, eigenvalues arise only for eigenstates.
Question 4 True / False
Eigenstates of a Hermitian operator corresponding to different eigenvalues are mutually orthogonal.
TTrue
FFalse
Answer: True
This is the orthogonality theorem for Hermitian operators: if Âψ_a = aψ_a and Âψ_b = bψ_b with a ≠ b, then ⟨ψ_a | ψ_b⟩ = 0. The proof follows directly from the Hermitian property. This orthogonality is physically meaningful: eigenstates with different eigenvalues represent mutually exclusive measurement outcomes. It also ensures that eigenstates form an orthogonal basis for the Hilbert space, so any state can be uniquely expanded as a sum of eigenstates with squared coefficients giving the probability distribution for measurement outcomes.
Question 5 Short Answer
Why must physical observables in quantum mechanics be represented by Hermitian operators? What two properties of Hermitian operators make them physically appropriate?
Think about your answer, then reveal below.
Model answer: Two properties: (1) Real eigenvalues — measurements always yield real numbers, so the operator representing an observable must have real eigenvalues; Hermitian operators ( = †) guarantee this. (2) Orthogonal eigenstates — eigenstates with different eigenvalues are orthogonal, forming a basis for the space of states. This means any state can be decomposed into eigenstates with coefficients whose squares give probabilities, making the Born rule well-defined. A non-Hermitian operator could have complex eigenvalues or non-orthogonal eigenstates, both of which are unphysical for a measurable observable.
The Hermitian requirement is not just mathematical convenience — it is physically necessary. Real eigenvalues match the reality of what measuring instruments register. Orthogonal eigenstates allow a consistent probability interpretation: if outcomes A and B are distinct measurement results, the corresponding states must be orthogonal so their probabilities add correctly without interference. The entire probability structure of quantum mechanics depends on this orthogonality. Hermiticity is what makes the Hilbert space formalism match the physical requirements of measurement.