Questions: Spin-1/2 Systems

Question 1

An electron is prepared in the eigenstate |↑⟩ of S_z (spin-up along z). The component S_x is then measured. What is the outcome?

• A. The result is always +ℏ/2, because spin-up means the spin vector points in the +z direction and has a positive x-component
• B. The measurement yields +ℏ/2 or −ℏ/2 with equal probability, because |↑⟩ is an equal superposition of the S_x eigenstates
• C. The measurement is undefined, because a spin-½ state along z has no x-component
• D. The result depends on the magnitude of the total spin angular momentum ℏ√(3)/2

Answer: (B)

The S_z eigenstates |↑⟩ and |↓⟩ are equal superpositions of the S_x eigenstates (|+x⟩ and |−x⟩). Measuring S_x on |↑⟩ collapses it to one of those eigenstates with probability 1/2 each. The misconception in option A is treating the quantum spin vector as a classical arrow pointing in a definite direction — in quantum mechanics, knowing the z-component tells you nothing definite about x or y, because the operators don't commute: [S_x, S_z] ≠ 0.

Question 2

What does a point on the Bloch sphere represent in the context of spin-½ quantum mechanics?

• A. The classical trajectory of the spinning electron as it orbits the nucleus
• B. A specific energy eigenstate of the hydrogen atom
• C. A unique pure quantum state of the two-level system, with north/south poles as spin-up/down along z and equatorial points as equal superpositions
• D. The probability density for finding the electron's spin in a given orientation, integrated over all measurements

Answer: (C)

The Bloch sphere is a geometric representation of all normalized pure states of a two-level (spin-½) system. Every point on the unit sphere corresponds to a spinor α|↑⟩ + β|↓⟩ with |α|² + |β|² = 1. The poles are the S_z eigenstates; equatorial points are equal superpositions differing in relative phase. This is a purely quantum-mechanical object — there is no classical analogue for a two-state system with continuously variable phases.

Question 3

The spin operators S_x, S_y, and S_z for a spin-½ particle obey the same commutation relations as orbital angular momentum: [S_x, S_y] = iℏS_z and its cyclic permutations.

• T. True
• F. False

Answer: True

Spin angular momentum is defined precisely by these commutation relations — they are the algebraic definition of angular momentum in quantum mechanics, satisfied by both orbital and spin operators. The remarkable feature of spin-½ is that this algebra is realized in a 2-dimensional Hilbert space with no spatial wavefunction, purely through the Pauli matrices. This is what makes spin-½ an 'intrinsic' angular momentum not reducible to orbital motion.

Question 4

An electron in the eigenstate |↑⟩ of S_z has simultaneously definite values for S_x and S_y, since |↑⟩ is a fully specified quantum state.

• T. True
• F. False

Answer: False

A fully specified quantum state does not mean all observables are definite — only observables that commute with the state's eigenvalue equation. Since [S_x, S_z] ≠ 0 and [S_y, S_z] ≠ 0, the S_x and S_y components are fundamentally uncertain in the state |↑⟩. Measuring S_x on |↑⟩ gives +ℏ/2 or −ℏ/2 with equal probability. The quantum uncertainty principle applies: you can know one component of spin precisely only at the cost of complete uncertainty in the perpendicular components.

Question 5

Why is the spin-½ system called the 'minimal nontrivial quantum system,' and why is it so central to quantum mechanics?

Model answer: A spin-½ system has a 2-dimensional Hilbert space — the smallest dimension that is still nontrivial (a 1-dimensional space would have only one state and no superposition). With just two basis states and a 2×2 matrix algebra, the spin-½ system demonstrates every characteristic feature of quantum mechanics: superposition, non-commuting observables, measurement-induced collapse, and entanglement (when two spin-½ particles are combined). It is the proving ground for quantum postulates precisely because it is simple enough to be fully solvable yet complex enough to exhibit genuine quantum behavior.

Beyond its pedagogical role, spin-½ is physically central: every electron, proton, and neutron is a spin-½ particle. Atomic structure, the Pauli exclusion principle, the periodic table, chemical bonding, and MRI all depend on it. The Pauli matrices that represent spin-½ operators appear throughout quantum field theory. It is the prototype for all two-level quantum systems — photon polarization, qubits in quantum computing — making spin-½ the most important 'toy model' that is also a real system.