An electron has spin quantum number s = ½. What is the magnitude of its spin angular momentum vector?
Aℏ/2, since the spin quantum number directly gives the angular momentum in units of ℏ
Bℏ√(3)/2, since the magnitude is ℏ√(s(s+1)) = ℏ√(3/4)
Cℏ, since the two projections +½ and −½ together account for a total angular momentum of 1
DZero, since spin has no spatial extent and therefore no angular momentum
The magnitude of any quantum angular momentum is ℏ√(j(j+1)), not ℏj. For s = ½, this gives ℏ√(½ · 3/2) = ℏ√(3)/2. The value ±ℏ/2 is only the z-component (projection) of the spin vector — what you measure along a chosen axis. The spin vector is never fully aligned with any axis; its total magnitude ℏ√(3)/2 is always larger than either projection ±ℏ/2. Confusing the projection with the magnitude is the most common error in this topic.
Question 2 Multiple Choice
In the Stern–Gerlach experiment, a beam of silver atoms splits into exactly two spots in an inhomogeneous magnetic field. Why does this result require a half-integer angular momentum quantum number?
AHalf-integer quantum numbers only apply to intrinsic properties, while integer values apply to orbital motion
BAn angular momentum quantum number l produces 2l+1 projection values; to get exactly 2 projections requires l = ½
CThe magnetic field was too weak to split a beam into more than two parts, regardless of the quantum number
DSilver atoms are electrically neutral, and neutral particles can only have two spin states
For orbital angular momentum with integer l, the number of allowed m projections is 2l+1: l=0 gives 1, l=1 gives 3, l=2 gives 5, and so on — always an odd number. There is no integer l that gives exactly 2. To produce exactly two projection values, you need 2s+1 = 2, which requires s = ½. This is precisely why the Stern–Gerlach result was inexplicable with existing orbital quantum numbers and required the postulate of a new half-integer quantum number — spin.
Question 3 True / False
Electron spin can be understood physically as the electron rotating about its own axis, analogous to Earth spinning on its axis.
TTrue
FFalse
Answer: False
This classical picture leads to irreconcilable contradictions. If you try to model spin as a literal rotation of a charged sphere, you can calculate the surface velocity needed to produce the observed magnetic moment — it exceeds the speed of light. Furthermore, the electron has no known spatial extent at the scale relevant to spin. Spin is a purely quantum mechanical, relativistic property with no classical analog; it emerges from the Dirac equation as an intrinsic feature of relativistic quantum fields. The word 'spin' is historical, not descriptive of physical rotation.
Question 4 True / False
Two electrons can occupy the same orbital (same n, l, m_l quantum numbers) provided they have opposite spin projections (m_s = +½ and m_s = −½).
TTrue
FFalse
Answer: True
The Pauli exclusion principle forbids two electrons from sharing all four quantum numbers. If two electrons share n, l, and m_l (same orbital), they must differ in m_s. Since m_s can only be +½ or −½, an orbital can hold at most two electrons — one spin-up and one spin-down. This is why spin is called the 'fourth quantum number' and why the electron's two-valued spin directly determines the shell-filling structure of the periodic table.
Question 5 Short Answer
Why can't electron spin be explained as a classical rotation, and what does 'spin ½' actually mean mathematically?
Think about your answer, then reveal below.
Model answer: Treating spin as literal rotation fails because the required surface velocity would exceed the speed of light, and the electron has no measurable spatial extent at the relevant scale — there is no surface to rotate. 'Spin ½' means the quantum number s = ½ is a fixed intrinsic property of all electrons. It determines the magnitude of the spin angular momentum vector, ℏ√(s(s+1)) = ℏ√(3)/2, and restricts the allowed projections onto any measurement axis to exactly two eigenvalues: m_s = +½ or m_s = −½, corresponding to +ℏ/2 and −ℏ/2 respectively. These are eigenvalues of the spin projection operator, not descriptions of physical rotation.
The key distinction is between the quantum number s (which sets the magnitude of the spin vector) and the projection m_s (which is what any single measurement along a chosen axis will yield). The full spin vector has magnitude ℏ√(3)/2 and points in some direction in 'spin space,' but when you measure its component along any axis, you always get one of the two eigenvalues ±ℏ/2 — never any intermediate value. This two-state discreteness is the hallmark of a spin-½ particle.