In quantum mechanics, a singlet state usually refers to a system in which all electrons are paired. The term "singlet" originally meant a linked set of particles whose net angular momentum is zero, that is, whose overall spin quantum number s = 0 {\displaystyle s=0} . As a result, there is only one spectral line of a singlet state. In contrast, a doublet state contains one unpaired electron and shows splitting of spectral lines into a doublet, and a triplet state has two unpaired electrons and shows threefold splitting of spectral lines.
History Singlets and the related spin concepts of doublets and triplets occur frequently in atomic physics and nuclear physics, where one often needs to determine the total spin of a collection of particles. Since the only observed fundamental particle with zero spin is the extremely inaccessible Higgs boson, singlets in everyday physics are necessarily composed of sets of particles whose individual spins are non-zero, e.g. 1/2 or 1. The origin of the term "singlet" is that bound quantum systems with zero net angular momentum emit photons within a single spectral line, as opposed to double lines (doublet state) or triple lines (triplet state). The number of spectral lines n {\displaystyle n} in this singlet-style terminology has a simple relationship to the spin quantum number: n = 2 s + 1 {\displaystyle n=2s+1} , and s = ( n − 1 ) / 2 {\displaystyle s=(n-1)/2} . Singlet-style terminology is also used for systems whose mathematical properties are similar or identical to angular momentum spin states, even when traditional spin is not involved. In particular, the concept of isospin was developed early in the history of particle physics to address the remarkable similarities of protons and neutrons. Within atomic nuclei, protons and neutrons behave in many ways as if they were a single type of particle, the nucleon, with two states. The proton-neutron pair thus by analogy was referred to as a doublet, and the hypothesized underlying nucleon was assigned a spin-like doublet quantum number I 3 = 1 2 {\displaystyle I_{3}={\tfrac {1}{2}}} to differentiate between those two states. Thus the neutron became a nucleon with isospin I 3 ( n ) = − 1 2 {\displaystyle I_{3}(n)=-{\tfrac {1}{2}}} , and the proton a nucleon with I 3 ( p ) = + 1 2 {\displaystyle I_{3}(p)=+{\tfrac {1}{2}}} . The isospin doublet notably shares the same SU(2) mathematical structure as the s = 1 2 {\displaystyle s={\tfrac {1}{2}}} angular momentum doublet. It should be mentioned that this early particle physics focus on nucleons was subsequently replaced by the more fundamental quark model, in which protons and neutrons are interpreted as bound systems of three quarks each. The isospin analogy also applies to quarks, and is the source of the names up (as in "isospin up") and down (as in "isospin down") for the quarks found in protons and neutrons. While for angular momentum states the singlet-style terminology is seldom used beyond triplets (spin=1), it has proven historically useful for describing much larger particle groups and subgroups that share certain features and are distinguished from each other by quantum numbers beyond spin. An example of this broader use of singlet-style terminology is the nine-member "nonet" of the pseudoscalar mesons.
Examples The simplest possible angular momentum singlet is a set (bound or unbound) of two spin-1/2 (fermion) particles that are oriented so that their spin directions ("up" and "down") oppose each other; that is, they are antiparallel. The simplest possible bound particle pair capable of exhibiting the singlet state is positronium, which consists of an electron and positron (antielectron) bound by their opposite electric charges. The electron and positron in positronium can also have identical or parallel spin orientations, which results in an experimentally-distinct form of positronium with a spin 1 or triplet state. An unbound singlet consists of a pair of entities small enough to exhibit quantum behavior (e.g. particles, atoms, or small molecules), not necessarily of the same type, for which four conditions hold:
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