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| A pair of [[spin (physics)|spin]]-1/2 particles can be combined to form one of three [[eigenstate|states]] of total spin 1 called the [[triplet state|triplet]], or a state of spin 0 which is called the singlet.<ref>[[David Griffiths (physicist)|D. J. Griffiths]], Introduction to Quantum Mechanics, Prentice Hall, Inc., 1995, pg. 165.</ref> In [[theoretical physics]], a '''singlet''' usually refers to a one-dimensional representation (e.g. a particle with vanishing [[spin (physics)|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.
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| A single electron has spin 1/2, and upon rotation its state transforms as a [[doublet state|doublet]], that is, as the [[fundamental representation]] of the [[Lie group]] [[SU(2)]].<ref>[[J. J. Sakurai]], Modern Quantum Mechanics, Addison Wesley, 1985.</ref> We can measure the spin of this electron's state by applying an operator <math>\vec{S}^2</math> to the state, and we will always obtain <math>\hbar^2 \, (1/2) \, (1/2 + 1) = (3/4) \, \hbar^2</math> (or spin 1/2) since the spin-up and spin-down states are both [[eigenstate]]s of this operator with the same eigenvalue.
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| Likewise, if we have a system of two electrons, we can measure the total spin by applying <math>\left(\vec{S}_1 + \vec{S}_2\right)^2</math>, where <math>\vec{S}_1</math> acts on electron 1 and <math>\vec{S}_2</math> 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 state|triplet]], containing three possible eigenstates.
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| In more mathematical language, we say the [[tensor product|product]] of two doublet representations can be decomposed into the sum of the [[Adjoint representation of a Lie group|adjoint representation]] (the [[triplet state|triplet]]) and the [[trivial representation]], the singlet.
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| 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 [[Bra-ket notation|Dirac notation]] this EPR state is usually represented as:
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| :<math>\frac{1}{\sqrt{2}}\left( \left|\uparrow \downarrow \right\rangle - \left|\downarrow \uparrow \right\rangle\right)</math>
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| == References ==
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| {{reflist}}
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| ==See also==
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| * [[Doublet state]]
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| * [[Triplet state]]
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| * [[Spin multiplicity]]
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| {{DEFAULTSORT:Singlet State}}
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| [[Category:Quantum mechanics]]
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| [[Category:Rotational symmetry]]
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| {{Quantum-stub}}
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Andrew Simcox is the name his mothers and fathers gave him and he totally loves this title. Invoicing is what I do for a living but I've always needed my personal business. One of the issues she enjoys most is canoeing and she's been performing it for fairly a while. Some time ago he selected to live in North Carolina and he doesn't plan on changing it.
My web site: tarot readings