Conformal equivalence

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 ${\vec {S}}^{2}$ to the state, and we will always obtain $\hbar ^{2}\,(1/2)\,(1/2+1)=(3/4)\,\hbar ^{2}$ (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 $\left({\vec {S}}_{1}+{\vec {S}}_{2}\right)^{2}$, where ${\vec {S}}_{1}$ acts on electron 1 and ${\vec {S}}_{2}$ 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:

${\frac {1}{\sqrt {2}}}\left(\left|\uparrow \downarrow \right\rangle -\left|\downarrow \uparrow \right\rangle \right)$

References

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See also

• Doublet state
• Triplet state
• Spin multiplicity

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