Triplet state - Citizendia

In physics, spin is the angular momentum intrinsic to a body, as opposed to orbital angular momentum, which is the motion of its center of mass about an external point. Physics (Greek Physis - φύσις in everyday terms is the Science of Matter and its motion. In Quantum mechanics, spin is a fundamental property of atomic nuclei, Hadrons and Elementary particles For particles with non-zero spin In Physics, the angular momentum of a particle about an origin is a vector quantity equal to the mass of the particle multiplied by the Cross product of the position The Azimuthal quantum number (or orbital angular momentum quantum number, second quantum number) symbolized as l (lower-case L is a Quantum number In quantum mechanics, spin is particularly important for systems at atomic length scales, such as individual atoms, protons, or electrons. Quantum mechanics is the study of mechanical systems whose dimensions are close to the Atomic scale such as Molecules Atoms Electrons History See also Atomic theory, Atomism The concept that matter is composed of discrete units and cannot be divided into arbitrarily tiny The proton ( Greek πρῶτον / proton "first" is a Subatomic particle with an Electric charge of one positive The electron is a fundamental Subatomic particle that was identified and assigned the negative charge in 1897 by J Such particles and the spin of quantum mechanical systems ("particle spin") possesses several unusual or non-classical features, and for such systems, spin angular momentum cannot be associated with rotation but instead refers only to the presence of angular momentum. 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 total spin angular momentum of one (in units of $\hbar$ ).

1 Two Spin 1/2 Particles
2 See also
3 References

Two Spin 1/2 Particles

In a system with two spin 1/2 particles - for example the proton and electron in the ground state of hydrogen, measured on a given axis, each particle can be either spin up or spin down so the system has four basis states in all

$\uparrow\uparrow,\uparrow\downarrow,\downarrow\uparrow,\downarrow\downarrow$

using the single particle spins to label the basis states, where the first and second arrow in each combination indicate the spin direction of the first and second particle respectively.

More rigorously

$|s_1,m_1\rangle|s_2,m_2\rangle=|s_1,m_1\rangle\otimes|s_2,m_2\rangle$

and since for spin-1/2 particles, the $|1/2,m\rangle$ basis states span a 2-dimensional space. Therefore the $|1/2,m_1\rangle|1/2,m_2\rangle$ basis states span a 4-dimensional space.

Now the total spin and its projection onto the previously defined axis can be computed using the rules for adding angular momentum in quantum mechanics using the Clebsch-Gordan coefficients. Quantum mechanics is the study of mechanical systems whose dimensions are close to the Atomic scale such as Molecules Atoms Electrons In Physics, the Clebsch-Gordan coefficients are sets of numbers that arise in Angular momentum coupling under the laws of Quantum mechanics. In general

$|j,m\rangle = \sum_{m_1+m_2=m}C_{m_1m_2m}^{s_1s_2s}|s_1m_1\rangle|s_2m_2\rangle$

substituting in the four basis states

$|1/2,+1/2\rangle|1/2,+1/2\rangle\ (\uparrow\uparrow)$

$|1/2,+1/2\rangle|1/2,-1/2\rangle\ (\uparrow\downarrow)$

$|1/2,-1/2\rangle|1/2,+1/2\rangle\ (\downarrow\uparrow)$

$|1/2,-1/2\rangle|1/2,-1/2\rangle\ (\downarrow\downarrow)$

returns the possible values for total spin given along with their representation in the $|1/2\ m_1\rangle|1/2\ m_2\rangle$ basis. There are three states with total angular momentum 1

$\left( \begin{array}{ll} |1,1\rangle & =\uparrow\uparrow\\ |1,0\rangle & =\frac{1}{\sqrt{2}}(\uparrow\downarrow + \downarrow\uparrow)\\ |1,-1\rangle & =\downarrow\downarrow \end{array} \right)\ s=1\ (\mathrm{triplet})$

and a fourth with total angular momentum 0

$\left(|0,0\rangle=\frac{1}{\sqrt{2}}(\uparrow\downarrow - \downarrow\uparrow)\right)\ s=0\ (\mathrm{singlet})$

The result is that a combination of two spin 1/2 particles can carry a total spin of 1 or 0, depending on whether they occupy a triplet or singlet state.

References

Griffiths, David J. In Theoretical physics, a singlet usually refers to a one-dimensional representation (e In Quantum mechanics, a doublet is a quantum state of a system with a spin of 1/2 such that there are two allowed values of the spin component −1/2 and +1/2 In Physics, the angular momentum of a particle about an origin is a vector quantity equal to the mass of the particle multiplied by the Cross product of the position The Pauli matrices are a set of 2 × 2 complex Hermitian and unitary matrices. In Quantum mechanics, spin multiplicity denotes the number of possible quantum states of a system with given principal spin quantum number S In Atomic physics, the spin quantum number is a Quantum number that parameterizes the intrinsic Angular momentum (or spin angular momentum or simply In Quantum mechanics, spin is an intrinsic property of all elementary particles. In Mathematics and Mathematical physics, the Euclidean group SE(d of Direct isometries is generated by Translations In Mathematics and Physics, in particular in the theory of the Orthogonal groups spinors are elements of a complex vector space introduced to expand the (2004). Introduction to Quantum Mechanics (2nd ed. ). Prentice Hall. ISBN 0-13-111892-7.
Shankar, R. (1994). "chapter 14-Spin", Principles of Quantum Mechanics (2nd ed. ). Springer. ISBN 0-306-44790-8.

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Contents

Two Spin 1/2 Particles

See also

References