Everything about Spin- totally explained
In
quantum mechanics,
spin is an intrinsic property of all
elementary particles.
Fermions, the particles that constitute ordinary matter, have
half-integer spin.
Spin-½ particles constitute an important subset of such fermions. All known elementary particles that are fermions have spin ½.
Overview
Particles having spin ½ include the
electron,
proton,
neutron,
neutrino, and
quarks. The dynamics of spin-½ objects can't be accurately described using
classical physics; they're 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
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.
Creation and annihilation 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), can't be measured simultaneously. Physically, this means that it's 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.
Symmetry
Mathematically, quantum mechanical spin isn't described by a
vector as in classical angular momentum. It is described using a family of objects known as
spinors. There are subtle differences between the behavior of spinors and vectors under
coordinate rotations. Rotating a spin-1/2 particle by 360 degrees doesn't bring it back to the same quantum state, but to the state with the opposite quantum
phase; this is detectable, in principle, with
interference experiments. To return the particle to its exact original state, one needs a 720 degree rotation!
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
effects a measurement of the spin in the
z direction.
»
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.
Further Information
Get more info on 'Spin-'.
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