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view source for spin physics wikipedia jump to content main menu main menu move to sidebar hide navigation main page contents current events random article about wikipedia contact us contribute help learn to edit community portal recent changes upload file special pages search search appearance donate create account log in personal tools donate create account log in view source for spin physics add languages article talk english read edit view history tools tools move to sidebar hide actions read edit view history general what links here related changes upload file page information get shortened url switch to legacy parser in other projects wikidata item appearance move to sidebar hide spin physics you do not have permission to edit this page for the following reason this ip address range has been globally blocked this does not affect your ability to read wikipedia pages most people who see this message have done nothing wrong some kinds of blocks restrict editing from specific service providers or telecom companies in response to recent abuse or vandalism and can sometimes affect other users who are unrelated to that abuse review the information below for assistance if you do not believe that you have done anything wrong this block affects editing on all wikimedia wikis the ip address or range 5 135 0 0 16 has been globally blocked by jon kolbert for the following reason s open proxy webhost visit the faq if you are affected this block will expire on 04 29 8 january 2027 your current ip address is 5 135 42 194 even while globally blocked you will usually still be able to edit pages on meta wiki if you believe you were blocked by mistake you can find additional information and instructions in the stewards block wizard other useful links global blocks help i have been blocked you can view and copy the source of this page higher spins see also 3d rotation group a note on lie algebras the spin sfrac 1 2 operator math 1 s sfrac ħ 2 σ forms the fundamental representation of representation theory of su 2 su 2 by taking kronecker product s of this representation with itself repeatedly one may construct all higher irreducible representations that is the resulting spin operator s for higher spin systems in three spatial dimensions can be calculated for arbitrarily large mvar s using this spin operator and ladder operator angular momentum ladder operators for example taking the kronecker product of two spin sfrac 1 2 yields a four dimensional representation which is separable into a 3 dimensional spin 1 triplet state s and a 1 dimensional spin 0 representation singlet state the resulting irreducible representations yield the following spin matrices and eigenvalues in the z basis ordered list for spin 1 they are math display block begin align s_x frac hbar sqrt 2 begin pmatrix 0 1 0 1 0 1 0 1 0 end pmatrix left 1 1 right rangle_x frac 1 2 begin pmatrix 1 sqrt 2 1 end pmatrix left 1 0 right rangle_x frac 1 sqrt 2 begin pmatrix 1 0 1 end pmatrix left 1 1 right rangle_x frac 1 2 begin pmatrix 1 sqrt 2 1 end pmatrix s_y frac hbar sqrt 2 begin pmatrix 0 i 0 i 0 i 0 i 0 end pmatrix left 1 1 right rangle_y frac 1 2 begin pmatrix 1 i sqrt 2 1 end pmatrix left 1 0 right rangle_y frac 1 sqrt 2 begin pmatrix 1 0 1 end pmatrix left 1 1 right rangle_y frac 1 2 begin pmatrix 1 i sqrt 2 1 end pmatrix s_z hbar begin pmatrix 1 0 0 0 0 0 0 0 1 end pmatrix left 1 1 right rangle_z begin pmatrix 1 0 0 end pmatrix left 1 0 right rangle_z begin pmatrix 0 1 0 end pmatrix left 1 1 right rangle_z begin pmatrix 0 0 1 end pmatrix end align math for spin sfrac 3 2 they are math display block begin array lclc s_x frac hbar2 begin pmatrix 0 sqrt 3 0 0 sqrt 3 0 2 0 0 2 0 sqrt 3 0 0 sqrt 3 0 end pmatrix left frac 3 2 frac 3 2 right rangle_x frac 1 2 sqrt 2 begin pmatrix 1 sqrt 3 sqrt 3 1 end pmatrix left frac 3 2 frac 1 2 right rangle_x frac 1 2 sqrt 2 begin pmatrix sqrt 3 1 1 sqrt 3 end pmatrix left frac 3 2 frac 1 2 right rangle_x frac 1 2 sqrt 2 begin pmatrix sqrt 3 1 1 sqrt 3 end pmatrix left frac 3 2 frac 3 2 right rangle_x frac 1 2 sqrt 2 begin pmatrix 1 sqrt 3 sqrt 3 1 end pmatrix s_y frac hbar2 begin pmatrix 0 i sqrt 3 0 0 i sqrt 3 0 2i 0 0 2i 0 i sqrt 3 0 0 i sqrt 3 0 end pmatrix left frac 3 2 frac 3 2 right rangle_y frac 1 2 sqrt 2 begin pmatrix i sqrt 3 i sqrt 3 1 end pmatrix left frac 3 2 frac 1 2 right rangle_y frac 1 2 sqrt 2 begin pmatrix i sqrt 3 1 i sqrt 3 end pmatrix left frac 3 2 frac 1 2 right rangle_y frac 1 2 sqrt 2 begin pmatrix i sqrt 3 1 i sqrt 3 end pmatrix left frac 3 2 frac 3 2 right rangle_y frac 1 2 sqrt 2 begin pmatrix i sqrt 3 i sqrt 3 1 end pmatrix s_z frac hbar2 begin pmatrix 3 0 0 0 0 1 0 0 0 0 1 0 0 0 0 3 end pmatrix left frac 3 2 frac 3 2 right rangle_z begin pmatrix 1 0 0 0 end pmatrix left frac 3 2 frac 1 2 right rangle_z begin pmatrix 0 1 0 0 end pmatrix left frac 3 2 frac 1 2 right rangle_z begin pmatrix 0 0 1 0 end pmatrix left frac 3 2 frac 3 2 right rangle_z begin pmatrix 0 0 0 1 end pmatrix end array math for spin sfrac 5 2 they are math display block begin align boldsymbol s _x frac hbar 2 begin pmatrix 0 sqrt 5 0 0 0 0 sqrt 5 0 2 sqrt 2 0 0 0 0 2 sqrt 2 0 3 0 0 0 0 3 0 2 sqrt 2 0 0 0 0 2 sqrt 2 0 sqrt 5 0 0 0 0 sqrt 5 0 end pmatrix boldsymbol s _y frac hbar 2 begin pmatrix 0 i sqrt 5 0 0 0 0 i sqrt 5 0 2i sqrt 2 0 0 0 0 2i sqrt 2 0 3i 0 0 0 0 3i 0 2i sqrt 2 0 0 0 0 2i sqrt 2 0 i sqrt 5 0 0 0 0 i sqrt 5 0 end pmatrix boldsymbol s _z frac hbar 2 begin pmatrix 5 0 0 0 0 0 0 3 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 3 0 0 0 0 0 0 5 end pmatrix end align math the generalization of these matrices for arbitrary spin mvar s is math display block begin align left s_x right _ ab frac hbar 2 left delta_ a b 1 delta_ a 1 b right sqrt s 1 a b 1 ab left s_y right _ ab frac i hbar 2 left delta_ a b 1 delta_ a 1 b right sqrt s 1 a b 1 ab left s_z right _ ab hbar s 1 a delta_ a b hbar s 1 b delta_ a b end align math where indices math a b math are integer numbers such that math display block 1 le a le 2s 1 quad 1 le b le 2s 1 math also useful in the quantum mechanics of multiparticle systems the general pauli group mvar g sub n sub is defined to consist of all mvar n fold tensor products of pauli matrices the analog formula of pauli matrices exponential of a pauli vector euler s formula in terms of the pauli matrices math display block hat r theta hat mathbf n e i frac theta 2 hat mathbf n cdot boldsymbol sigma i cos frac theta 2 i left hat mathbf n cdot boldsymbol sigma right sin frac theta 2 math for higher spins is tractable but less simple ref cite journal doi 10 3842 sigma 2014 084 last1 curtright last2 fairlie last3 zachos first1 t l first2 d b first3 c k author link thomas curtright author link2 david fairlie author link3 cosmas zachos year 2014 title a compact formula for rotations as spin matrix polynomials journal sigma volume 10 page 084 arxiv 1402 3541 bibcode 2014sigma 10 084c ref return to spin physics retrieved from https en wikipedia org wiki spin_ physics privacy 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