Spin — intrinsic angular momentum — Physics notes

WHY does spin exist? (the experimental forcing)

WHAT is spin? (definition)

For an electron $s=\tfrac12$ , so $m_s = \pm\tfrac12$ → exactly two states ("spin up" $\uparrow$ , "spin down" $\downarrow$ ). This is why Stern–Gerlach gave two spots.

HOW the numbers come out (derive, don't memorise)

Spin obeys the same algebra as any angular momentum. We take that as the first principle: $[S_x,S_y]=i\hbar S_z \quad(\text{and cyclic}).$

Why this matters: these commutators are the entire foundation. Everything below follows from them, exactly as for orbital $\vec L$ .

From the algebra, the operator $\vec S^2$ commutes with each component, so a state can have definite total magnitude. Its eigenvalue is $\vec S^2 \,\psi = \hbar^2\, s(s+1)\,\psi.$ Why $s(s+1)$ and not $s^2$ ? Because $\vec S^2 = S_x^2+S_y^2+S_z^2$ ; the "extra" $+s$ is the contribution of the components $S_x,S_y$ that can never simultaneously be zero (uncertainty). Hence $|\vec S| = \hbar\sqrt{s(s+1)}.$

Step 2 — projection

$S_z$ is quantized in steps of $\hbar$ : $S_z = m_s\hbar,\qquad m_s=-s,-s+1,\dots,+s.$ Why these run from $-s$ to $+s$ ? Ladder operators $S_\pm=S_x\pm iS_y$ raise/lower $m_s$ by 1; the ladder must terminate at top and bottom, forcing the range to be symmetric and $2s$ to be an integer. For $s=\tfrac12$ the only allowed values are $m_s=\pm\tfrac12$ .

Step 3 — the magnetic moment

A charged particle with angular momentum has a magnetic moment. For spin: $\boxed{\;\vec\mu_s = -g_s\,\frac{e}{2m_e}\,\vec S\;}$ Why the extra factor $g_s$ ? Classically a current loop gives $\mu = \tfrac{e}{2m}L$ (the " $g=1$ " guess). Experiment + Dirac's relativistic theory show the electron's spin is twice as magnetic: $g_s \approx 2$ . This anomalous " $g=2$ " is a deep, purely quantum result.

Worked examples

Forecast-then-Verify

Common mistakes (steel-manned)

Flashcards

What is spin?

An intrinsic, fixed angular momentum of a particle, unrelated to spatial motion, quantized with possibly half-integer quantum number.

Spin quantum number of an electron

s=\tfrac12

Allowed

m_s

values for an electron

+\tfrac12

and

-\tfrac12

(two states).

Magnitude of electron spin

|\vec S|=\hbar\sqrt{s(s+1)}=\frac{\sqrt3}{2}\hbar

Why did Stern–Gerlach give two spots?

Because

m_s=\pm\tfrac12

→ exactly two orientations of the magnetic moment.

Number of Stern–Gerlach spots for spin

s

2s+1

Spin g-factor of the electron

g_s\approx 2

(anomalous; not 1).

Bohr magneton expression

\mu_B=\dfrac{e\hbar}{2m_e}

, the magnitude of electron's spin

z

-moment.

Difference between

S_z

and

|\vec S|

S_z=m_s\hbar

is the projection;

|\vec S|=\hbar\sqrt{s(s+1)}

is the full magnitude (larger).

Angle of electron spin from z-axis

\theta=\arccos\!\frac{1}{\sqrt3}\approx54.7^\circ

Force on a spin in a field gradient

F_z=\mu_z\,\partial B_z/\partial z

, with

\mu_z=\mp\mu_B

Recall Feynman: explain to a 12-year-old

Imagine every electron comes from the factory with a tiny built-in "twirl." You can't make it twirl more or less — it always twirls the same amount. When you put it near a magnet, it can only point its twirl in two ways: a little up, or a little down. That's why a beam of atoms splits neatly into two — not a smear, just two. The twirl isn't a real spinning ball; it's a rule of the quantum world. We call this built-in twirl spin.

Connections

Stern-Gerlach Experiment — the experiment that revealed spin.
Orbital Angular Momentum — same algebra, integer-only quantum numbers.
Quantum Numbers — $n,\ell,m_\ell,m_s$ and the full state label.
Pauli Exclusion Principle — relies on spin- $\tfrac12$ (fermions).
Zeeman Effect & Fine Structure — energy splitting from $U=-\vec\mu\cdot\vec B$ .
Bohr Magneton — natural unit of magnetic moment.
Dirac Equation — explains $g_s=2$ from first principles.

Concept Map

Hinglish (regional understanding)

Intuition Hinglish mein samjho

Dekho, spin ka matlab yeh nahi hai ki electron koi chhoti gend ki tarah ghoom raha hai. Spin ek intrinsic (andar se built-in) angular momentum hai — jaise electron ka charge aur mass fixed hota hai, waise hi uska spin bhi fixed hota hai. Tum isse na badha sakte ho, na ghata sakte ho. Iska magnitude hamesha $|\vec S|=\frac{\sqrt3}{2}\hbar$ rehta hai.

Yeh idea aaya kahaan se? Stern–Gerlach experiment se. Silver atoms ka beam ek magnet ke gradient se guzra, aur expectation thi ek hi spot, lekin mile do spots! Matlab andar koi cheez hai jiske sirf do orientations possible hain — spin up ( $m_s=+\tfrac12$ ) aur spin down ( $m_s=-\tfrac12$ ). Isi se pata chala ki electron ka spin quantum number $s=\tfrac12$ hai. General rule: $2s+1$ spots milte hain.

Do important baatein yaad rakho. Pehli: projection aur magnitude alag hain. $S_z=m_s\hbar=\tfrac12\hbar$ projection hai, par poora vector $|\vec S|=\frac{\sqrt3}{2}\hbar$ hota hai — thoda zyada, kyunki $S_x$ aur $S_y$ kabhi exactly zero nahi ho sakte (uncertainty principle). Isiliye spin vector kabhi seedha $z$ -axis pe align nahi hota, hamesha $54.7^\circ$ ka angle banata hai.

Doosri baat: spin ka magnetic moment classical formula se double strong hai — yani $g_s\approx 2$ . Yeh ek deep quantum/relativistic result hai (Dirac equation se aata hai). Exam me yaad rakhna: orbital ke liye $g=1$ , par spin ke liye $g\approx 2$ . Yeh chhoti si galti bahut common hai, isse bacho!

Spin — intrinsic angular momentum

WHY does spin exist? (the experimental forcing)

WHAT is spin? (definition)

HOW the numbers come out (derive, don't memorise)

Step 1 — magnitude

Step 2 — projection

Step 3 — the magnetic moment

Worked examples

Forecast-then-Verify

Common mistakes (steel-manned)

Flashcards

Connections

Concept Map

Hinglish (regional understanding)

Go deeper — visual, from zero

Test yourself — Modern Physics

Connections