Spin-½

In quantum mechanics, spin is an intrinsic property of all elementary particles. Fermions, the particles that make up ordinary matter, have half-integer spin. An important special case of this are the spin-½ particles. All known elementary particles that are fermions have spin ½.

Overview

Particles which have spin ½ include the electron, the proton, the neutron, the neutrino, and the quarks. Spin-½ objects have peculiar dynamics which are not accurately described using classical physics; they are among the simplest systems which require quantum mechanics to describe them. As such, the study of the behavior of spin-½ systems forms a central part of undergraduate and graduate-level instruction in quantum mechanics.

General properties

Spin-½ objects are all Fermions, a fact explained by the spin statistics theorem, and satisfy the Pauli exclusion principle. Spin-½ particles can have a permanent magnetic moment along the direction of their spin, and this magnetic moment gives rise to electromagnetic interactions that depend on the spin. One such effect that was important in the discovery of spin is the Zeeman effect.

Unlike in more complicated quantum mechanical systems, the spin of a spin-½ particle can be expressed as a linear combination of just two eigenstates, or eigenspinors. These are traditionally labeled spin up and spin down. Because of this the quantum mechanical spin operators can be represented as simple 2 × 2 matrices, as opposed to the infinite dimensional matrices commonly needed to represent operators like energy or position. These matrices are called the Pauli matrices.

Raising and lowering operators can be constructed for spin-½ objects; these obey the same commutation relations as other angular momentum operators.

Connection to the uncertainty principle

One consequence of the generalized uncertainty principle is that the spin projection operators (which measure the spin along a given direction like x, y, or z), cannot be measured simultaneously. Physically, this means that it is ill defined what axis a particle is spinning about. A measurement of the z-component of spin destroys any information about the x and y components that might previously have been obtained.

Stern–Gerlach experiment

When a spin-½ particle with non-zero magnetic moment like an electron is placed in an inhomogenous magnetic field, it experiences a force. This acts to separate out particles in the spin up state from particles in the spin down state. This is the idea behind the Stern–Gerlach experiment.

Mathematical description

The quantum state of the spin of a spin-½ particle can be described by a complex-valued vector with two components called a two-component spinor. When spinors are used to describe the quantum states, quantum mechanical operators are represented by 2 × 2, complex-valued Hermitian matrices.

For example, the spin projection operator $S_z$ effects a measurement of the spin in the z direction.

S_z = \textstyle\frac{\hbar}{2} \sigma _z  = \textstyle\frac{\hbar}{2}  \begin{pmatrix}

1&0\\ 0&-1 \end{pmatrix}

$S_z$ operator has two eigenvalues, of $\pm \textstyle\frac{\hbar}{2}$ , which correspond to the eigenvectors

\begin{bmatrix} 1 \\ 0 \end{bmatrix} = \left \vert {s_z = +\textstyle\frac 1 2} \right \rangle = | {\uparrow} \rangle

\begin{bmatrix} 0 \\ 1 \end{bmatrix} = \left \vert {s_z = -\textstyle\frac 1 2} \right \rang  = | {\downarrow} \rang

These vectors form a complete basis for the Hilbert space describing the spin-½ particle. Thus, linear combinations of these two states can represent all possible states of the spin.

References

Griffiths, David J. (2005) Introduction to Quantum Mechanics (2nd ed.). Upper Saddle River, NJ: Pearson Prentice Hall. ISBN 0-13-111892-7.

Quantum mechanics | Rotational symmetry

自旋1/2