Spin Examples

Spin examples are finite-dimensional, but they are not just matrix exercises. The physical content is tied to measurement axes, rotations, tensor products, and representation conventions.

Example Routes

Need	Learn	Main Check
First spin-$1/2$ model	Spin-One-Half First Encounter	Which axis defines the basis?
Spinor conventions	Spin-Half Hilbert Space	Normalization and phase
Pauli algebra	Pauli Matrices	Sign and basis conventions
Rotation of a spinor	Spin Rotations	$SU(2)$ versus $SO(3)$ angle
Two spin-$1/2$ particles	Two Spin-Half Particles	Product basis order
Clebsch-Gordan coefficients	Clebsch-Gordan Coefficients	Phase convention
Compact formula	Spin-Half Matrices	$S_i=(\hbar/2)\sigma_i$

Minimal Worked Check

In the $S_z$ basis,

$$\lvert +x\rangle =\frac{1}{\sqrt2} \begin{pmatrix} 1\\ 1 \end{pmatrix}, \qquad \lvert +z\rangle = \begin{pmatrix} 1\\ 0 \end{pmatrix}.$$

The probability that a spin prepared in $\lvert +z\rangle$ is found up along $\hat x$ is

$$\lvert\langle +x\vert +z\rangle\rvert^2 =\frac12.$$

The number $1/2$ is not a statement of ignorance about a preexisting $S_x$ value; it is the Born-rule probability for a different measurement basis.

Common Mistakes

• Treating a spinor’s global phase as observable.
• Forgetting the factor of $\hbar/2$ between $\sigma_i$ and $S_i$.
• Confusing a spatial rotation angle with the spinor phase accumulated under $SU(2)$.
• Changing product-basis order halfway through a two-spin calculation.
• Comparing Clebsch-Gordan coefficients across sources without checking phase conventions.

Related Experiment Pages

• Stern–Gerlach Experiment gives the historical spin-measurement anchor.
• Bell Tests connect spin correlations to foundational inequalities.

References

• J. J. Sakurai and J. Napolitano, Modern Quantum Mechanics, 3rd ed., Cambridge University Press, 2020.
• R. Shankar, Principles of Quantum Mechanics, 2nd ed., Springer, 1994.
• D. J. Griffiths and D. F. Schroeter, Introduction to Quantum Mechanics, 3rd ed., Cambridge University Press, 2018.

Exercises

1. In the minimal worked check, what is the probability of finding spin down along $\hat x$?

Solution

The state $\lvert -x\rangle=(\lvert +z\rangle-\lvert -z\rangle)/\sqrt2$, so $\lvert\langle -x\vert +z\rangle\rvert^2=1/2$. The two outcomes along $\hat x$ exhaust the probabilities.