Silver atoms have one outer electron in an l = 0 orbital. In the Stern-Gerlach experiment, a beam of silver atoms was deflected into exactly two discrete spots. Why does this result require half-integer spin?
ATwo spots indicate that silver has two outer electrons, each contributing one unit of angular momentum
BSince l = 0 means no orbital angular momentum, the two spots must come from an intrinsic angular momentum with 2s + 1 = 2 possible z-projections, requiring s = 1/2 — integer spin would give an odd number of spots or no splitting
CTwo spots appear whenever a beam is split by a magnetic field gradient, regardless of the angular momentum of the atoms
DThe two spots correspond to spin-up and spin-down electrons, and any electron will produce exactly two spots regardless of its orbital state
The number of spots equals 2s + 1 (the number of distinct m_s values). With l = 0, orbital angular momentum cannot explain the splitting. Two spots means 2s + 1 = 2, so s = 1/2. An s = 1 particle would give 3 spots; s = 0 would give 1 (no splitting). The experiment rules out all integer values of s and proves that half-integer angular momentum is a physical reality — not a mathematical curiosity.
Question 2 Multiple Choice
Which statement best captures what it means to say an electron has spin s = 1/2?
AThe electron physically rotates about its own axis, and the rotation rate corresponds to half a full turn per unit time
BThe electron's intrinsic angular momentum is quantized, with z-component taking only values +ℏ/2 or −ℏ/2; this is a fundamental property with no classical rotating-object analog
CThe electron has half the angular momentum of a proton, which has spin 1
DThe electron orbit contributes angular momentum of ℏ/2 per revolution around the nucleus
Spin is not rotation. An electron is pointlike — it has no extended structure that could rotate. The 'spin' quantum number s = 1/2 is an intrinsic property like mass or charge, meaning it cannot be changed by any interaction. The mathematical structure is identical to orbital angular momentum (same commutator algebra), but the physical interpretation is fundamentally different: there is no classical picture of what is spinning.
Question 3 True / False
The spin quantum number s = 1/2 for an electron is a fixed, unchangeable property, similar to its mass and charge.
TTrue
FFalse
Answer: True
s is intrinsic — it characterizes the type of particle and cannot be altered by external fields, temperature, or any interaction. What can change is m_s (the spin projection along a chosen axis), which takes values +1/2 or −1/2. For example, a magnetic field or a measurement can flip an electron from spin-up to spin-down (changing m_s), but the value s = 1/2 is invariant.
Question 4 True / False
Because orbital angular momentum and spin obey the same commutator algebra [Ĵ_i, Ĵ_j] = iℏ ε_{ijk} Ĵ_k, spin should take the same integer values (l = 0, 1, 2, ...) as orbital angular momentum.
TTrue
FFalse
Answer: False
The commutator algebra alone allows any non-negative half-integer value (0, 1/2, 1, 3/2, ...). Orbital angular momentum is restricted to integers by an *additional* requirement: the spatial wavefunction must be single-valued under a 2π rotation (ψ must return to itself, not −ψ). Spin states live in a separate spin Hilbert space where no such single-valuedness constraint applies, so s = 1/2 is mathematically consistent. Nature uses this freedom: electrons, protons, and neutrons all have s = 1/2.
Question 5 Short Answer
Why is spin called 'intrinsic' angular momentum, and why is the picture of an electron physically spinning on its axis incorrect?
Think about your answer, then reveal below.
Model answer: Spin is 'intrinsic' because it is a permanent, unchangeable property of a particle — it does not arise from any motion or configuration and cannot be removed. The 'electron spinning' picture fails because the electron is a pointlike particle with no spatial extent, so there is nothing extended to rotate. Furthermore, if you tried to model the electron's observed magnetic moment as classical rotation, the surface of the electron would need to move faster than the speed of light. Spin is a purely quantum mechanical property with no classical analog.
This is one of the places where quantum mechanics simply cannot be understood by analogy to classical mechanics. The algebra of spin is the same as orbital angular momentum, which tempts people to imagine physical rotation. But the derivation of that algebra from commutation relations does not require any rotating object — it only requires the abstract structure of the operators. Spin is the clearest example of a quantum property that must be accepted on its own mathematical and experimental terms.