Questions: Quantum Superposition and Linear Combinations of States

Question 1

A student says: 'Saying an electron is in a superposition of spin-up and spin-down is just like saying a coin is spinning in the air — it's either heads or tails, we just don't know which yet.' What is the key reason this analogy fails?

• A. It fails because quantum objects cannot be compared to classical macroscopic objects
• B. It fails because quantum superposition enables interference between components — a classical probability mixture cannot produce interference patterns, but a genuine superposition can
• C. It fails because we can measure which spin-state the electron is in, whereas we cannot observe a coin mid-spin
• D. It fails because the coin analogy applies only to discrete systems, while quantum states are continuous

Answer: (B)

The critical distinction is interference. A genuine quantum superposition carries relative phases between its components that produce constructive and destructive interference in experiments (double-slit, interferometers). A classical 'we don't know which' probability mixture has no phases and cannot produce interference — the fringe pattern would disappear. The empirical signature of true quantum superposition is these interference effects, which depend on the phases of the coefficients cₙ, not just their magnitudes. Classical ignorance and quantum superposition make different experimental predictions.

Question 2

An electron is prepared as an equal superposition of spin-up and spin-down along the z-axis. Physicist A says it is 'in superposition.' Physicist B says it is 'in an eigenstate.' Are they contradicting each other?

• A. Yes — an eigenstate is by definition not a superposition, so both cannot be correct simultaneously
• B. No — the same state can be an eigenstate of one observable (spin-x) and a superposition of eigenstates of another (spin-z); whether a state 'is in superposition' depends entirely on the measurement basis
• C. Yes — only one correct decomposition of a quantum state exists at any time
• D. No — all quantum states are simultaneously eigenstates of every observable

Answer: (B)

An equal superposition of z-spin-up and z-spin-down is exactly the eigenstate of spin along the x-axis. Physicist A is measuring spin along z; physicist B is measuring spin along x. They are both correct, relative to their chosen basis. This is the deep insight: 'being in superposition' is not an intrinsic property of a state but a relationship between the state and a choice of observable. Every state is an eigenstate of some observable and a superposition of eigenstates of every non-commuting observable. Asking 'is this state in superposition?' without specifying the basis is not well-posed.

Question 3

Measurement collapses a quantum superposition, projecting the system onto one eigenstate and destroying the superposition.

• T. True
• F. False

Answer: True

Before measurement, the system evolves as a superposition ψ = Σcₙφₙ with each component carrying phase e^{-iEₙt/ℏ}. Measuring the corresponding observable instantaneously projects the system onto a single eigenstate φₙ with probability |cₙ|². The other components disappear — the superposition is destroyed. Subsequent measurements (before re-preparation) always return the same eigenstate, because the collapse has already selected it. To recover the original superposition, the system must be re-prepared from scratch.

Question 4

The probability of obtaining eigenstate φₙ when measuring a system in state ψ = Σcₙφₙ is given by the coefficient cₙ itself.

• T. True
• F. False

Answer: False

The probability is |cₙ|² — the squared modulus of the complex amplitude, not cₙ itself. The coefficients cₙ are complex numbers (probability amplitudes) with both magnitude and phase; they cannot directly be probabilities since they can be negative or imaginary. It is squaring the absolute value that yields a real, non-negative probability consistent with the normalization Σ|cₙ|² = 1. This distinction is physically crucial: the phases of cₙ drive quantum interference (which survives before measurement), while |cₙ|² gives the collapse probability (which discards the phase information).

Question 5

What is the key physical evidence that quantum superposition is fundamentally different from classical uncertainty, and what feature of the formalism produces this evidence?

Model answer: The key evidence is quantum interference — constructive and destructive fringe patterns in experiments like the double-slit or Mach-Zehnder interferometer. These patterns depend on the relative phases of the coefficients cₙ in ψ = Σcₙφₙ. A classical probability distribution over definite states has no phase information and cannot produce interference: if you blocked each path and added the resulting distributions, you would get a smooth sum, not fringes. Only a genuine superposition, where both amplitude and phase are present in each term, can cancel in some directions and reinforce in others. The phase is the fingerprint of true quantum superposition; interference is its observable consequence.

This is why physicists say quantum superposition is an ontological claim, not an epistemic one. It is not that we lack information about which state the system is in — the system genuinely is in multiple states simultaneously, and the phases between them have real physical consequences that can be measured. When we measure, those phases are lost (the collapse is irreversible); but before measurement, they are as real as any other physical quantity.