Singlet
Encyclopedia
A pair of spin
-1/2 particles can be combined to form one of three states of total spin 1 called the triplet
, or a state of spin 0 which is called the singlet. In theoretical physics
, a singlet usually refers to a one-dimensional representation (e.g. a particle with vanishing spin
). It may also refer to two or more particles prepared in a correlated state, such that the total angular momentum of the state is zero. Singlets and other such representations frequently occur in atomic physics
and nuclear physics
, where one tries to determine the total spin from a collection of particles.
A single electron has spin 1/2, and upon rotation its state transforms as a doublet, that is, as the fundamental representation
of the Lie group
SU(2). We can measure the spin of this electron's state by applying an operator
to the state, and we will always obtain 
(or spin 1/2) since the spin-up and spin-down states are both eigenstates of this operator with the same eigenvalue.
Likewise, if we have a system of two electrons, we can measure the total spin by applying
, where 
acts on electron 1 and 
acts on electron 2. However, we can now have two possible spins, which is to say, two possible eigenvalues of the total spin operator, corresponding to spin-0 or spin-1. Each eigenvalue belongs to a set of eigenstates. The "spin-0" set is called the singlet, containing one state (see below), and the "spin-1" set is called the triplet
, containing three possible eigenstates.
In more mathematical language, we say the product
of two doublet representations can be decomposed into the sum of the adjoint representation
(the triplet
) and the trivial representation
, the singlet.
The singlet state formed from a pair of electrons has many peculiar properties, and plays a fundamental role in the EPR paradox
and quantum entanglement
. In Dirac notation
this EPR state is usually represented as:
Spin (physics)
In quantum mechanics and particle physics, spin is a fundamental characteristic property of elementary particles, composite particles , and atomic nuclei.It is worth noting that the intrinsic property of subatomic particles called spin and discussed in this article, is related in some small ways,...
-1/2 particles can be combined to form one of three states of total spin 1 called the triplet
Triplet state
A spin triplet is a set of three quantum states of a system, each with total spin S = 1 . The system could consist of a single elementary massive spin 1 particle such as a W or Z boson, or be some multiparticle state with total spin angular momentum of one.In physics, spin is the angular momentum...
, or a state of spin 0 which is called the singlet. In theoretical physics
Theoretical physics
Theoretical physics is a branch of physics which employs mathematical models and abstractions of physics to rationalize, explain and predict natural phenomena...
, a singlet usually refers to a one-dimensional representation (e.g. a particle with vanishing spin
Spin (physics)
In quantum mechanics and particle physics, spin is a fundamental characteristic property of elementary particles, composite particles , and atomic nuclei.It is worth noting that the intrinsic property of subatomic particles called spin and discussed in this article, is related in some small ways,...
). It may also refer to two or more particles prepared in a correlated state, such that the total angular momentum of the state is zero. Singlets and other such representations frequently occur in atomic physics
Atomic physics
Atomic physics is the field of physics that studies atoms as an isolated system of electrons and an atomic nucleus. It is primarily concerned with the arrangement of electrons around the nucleus and...
and nuclear physics
Nuclear physics
Nuclear physics is the field of physics that studies the building blocks and interactions of atomic nuclei. The most commonly known applications of nuclear physics are nuclear power generation and nuclear weapons technology, but the research has provided application in many fields, including those...
, where one tries to determine the total spin from a collection of particles.
A single electron has spin 1/2, and upon rotation its state transforms as a doublet, that is, as the fundamental representation
Fundamental representation
In representation theory of Lie groups and Lie algebras, a fundamental representation is an irreducible finite-dimensional representation of a semisimple Lie group...
of the Lie group
Lie group
In mathematics, a Lie group is a group which is also a differentiable manifold, with the property that the group operations are compatible with the smooth structure...
SU(2). We can measure the spin of this electron's state by applying an operator
to the state, and we will always obtain (or spin 1/2) since the spin-up and spin-down states are both eigenstates of this operator with the same eigenvalue.Likewise, if we have a system of two electrons, we can measure the total spin by applying
, where acts on electron 1 and acts on electron 2. However, we can now have two possible spins, which is to say, two possible eigenvalues of the total spin operator, corresponding to spin-0 or spin-1. Each eigenvalue belongs to a set of eigenstates. The "spin-0" set is called the singlet, containing one state (see below), and the "spin-1" set is called the tripletTriplet state
A spin triplet is a set of three quantum states of a system, each with total spin S = 1 . The system could consist of a single elementary massive spin 1 particle such as a W or Z boson, or be some multiparticle state with total spin angular momentum of one.In physics, spin is the angular momentum...
, containing three possible eigenstates.
In more mathematical language, we say the product
Tensor product
In mathematics, the tensor product, denoted by ⊗, may be applied in different contexts to vectors, matrices, tensors, vector spaces, algebras, topological vector spaces, and modules, among many other structures or objects. In each case the significance of the symbol is the same: the most general...
of two doublet representations can be decomposed into the sum of the adjoint representation
Adjoint representation
In mathematics, the adjoint representation of a Lie group G is the natural representation of G on its own Lie algebra...
(the triplet
Triplet state
A spin triplet is a set of three quantum states of a system, each with total spin S = 1 . The system could consist of a single elementary massive spin 1 particle such as a W or Z boson, or be some multiparticle state with total spin angular momentum of one.In physics, spin is the angular momentum...
) and the trivial representation
Trivial representation
In the mathematical field of representation theory, a trivial representation is a representation of a group G on which all elements of G act as the identity mapping of V...
, the singlet.
The singlet state formed from a pair of electrons has many peculiar properties, and plays a fundamental role in the EPR paradox
EPR paradox
The EPR paradox is a topic in quantum physics and the philosophy of science concerning the measurement and description of microscopic systems by the methods of quantum physics...
and quantum entanglement
Quantum entanglement
Quantum entanglement occurs when electrons, molecules even as large as "buckyballs", photons, etc., interact physically and then become separated; the type of interaction is such that each resulting member of a pair is properly described by the same quantum mechanical description , which is...
. In Dirac notation
Bra-ket notation
Bra-ket notation is a standard notation for describing quantum states in the theory of quantum mechanics composed of angle brackets and vertical bars. It can also be used to denote abstract vectors and linear functionals in mathematics...
this EPR state is usually represented as:
See also
- Doublet state
- Triplet stateTriplet stateA spin triplet is a set of three quantum states of a system, each with total spin S = 1 . The system could consist of a single elementary massive spin 1 particle such as a W or Z boson, or be some multiparticle state with total spin angular momentum of one.In physics, spin is the angular momentum...
- Spin multiplicity