Spin

Quantized intrinsic angular momentum and its measurable projections. Theory, formulas, worked examples and interactive graphs.

Conceptual context

The $s$ is not the spin of a tiny rigid bead. It is an intrinsic quantum property revealed only through measurement. In an experiment, an axis is chosen, usually z, and one allowed $S_z$ appears. The central question is why the $m_s$ selects that measured result rather than a continuous range. The answer combines quantization, physical scaling through $\hbar$, and a reading rule: the system has a fixed $S$, but only certain projections on the measurement axis can be observed. With an external $B$, those projections also become measurable energy splittings.

🟢 Essential level

The $s$ fixes what kind of spin a particle or system has. For an electron, that value allows two answers when the spin is measured along a chosen axis: one aligned with the axis and one opposed to it.

The quantity that labels those answers is the $m_s$. It is not a hidden classical orientation, but a discrete label for the $S_z$ the apparatus can record. The $\hbar$ acts as the conversion scale from quantum label to physical angular momentum. Therefore the measured result does not slide through many intermediate values: it jumps between allowed projections. The $S$ stays fixed by the particle type, while the measurement selects one component. In an external magnetic field, each projection gives a different interaction energy. It is like flipping a coin that, instead of landing at any angle, could only ever show "heads" or "tails" once measured: the spin keeps a fixed total size, but each measurement along a given axis reveals only one of a few discrete marks, never an intermediate value.

🔵 Formal level

Basic formulas

The formal description separates the total size of spin from its measurable component along the z-axis. The $s$ determines the $S$ through the total-magnitude relation:

Magnitude of spin angular momentum

Here $\hbar$ supplies the physical unit of angular momentum. For a spin state defined relative to the z-axis, the $m_s$ directly determines the $S_z$:

Quantized spin projection

Real student questions

Does spin mean the electron rotates around itself? No. The $s$ describes intrinsic angular momentum, but not a rotating material sphere.

Why can I not measure any $S_z$ I want? Because $m_s$ allows only discrete values compatible with $s$.

Are $S$ and $S_z$ the same thing? No. The first is the total size; the second is a component along the chosen axis.

Why does $B$ enter? Because it converts differences in $\mu _z$ into observable differences in $E_z$.

Final synthesis

Spin shows that measurement does not reveal a continuous classical orientation. The $m_s$ selects a discrete $S_z$, scaled by $\hbar$, while $s$ preserves the total intrinsic angular momentum size.