SU(2) • the spin returns to its original direction after time In the two dimensional space t = 2π/ωc. {|Sz; ↑i, |Sz; ↓i} • the wave vector returns to its original value after time t = 4π/ωc. the spin operators ¯h Matrix representation Sx = {(|Sz; ↑ihSz; ↓ |) + (|Sz; ↓ihSz; ↑ |)} 2 In the basis {|Sz; ↑i, |Sz; ↓i} the base vectors are i¯h represented as S = {−(|S ; ↑ihS ; ↓ |) + (|S ; ↓ihS ; ↑ |)} y 2 z z z z 1 0 ¯h |S ; ↑i 7→ ≡ χ |S ; ↓i 7→ ≡ χ S = {(|S ; ↑ihS ; ↑ |) − (|S ; ↓ihS ; ↓ |)} z 0 ↑ z 1 ↓ z 2 z z z z † † hSz; ↑ | 7→ (1, 0) ≡ χ↑ hSz; ↓ | 7→ (0, 1) ≡ χ↓, satisfy the angular momentum commutation relations so an arbitrary state vector is represented as [Sx,Sy] = i¯hSz + cyclic permutations. hSz; ↑ |αi Thus the smallest dimension where these commutation |αi 7→ hSz; ↓ |αi relations can be realized is 2. hα| 7→ (hα|S ; ↑i, hα|S ; ↓i). The state z z The column vector |αi = |Sz; ↑ihSz; ↑ |αi + |Sz; ↓ihSz; ↓ |αi hS ; ↑ |αi c behaves in the rotation χ = z ≡ ↑ hSz; ↓ |αi c↓ iS φ D (φ) = exp − z z ¯h is called the two component spinor like Pauli’s spin matrices Pauli’s spin matrices σi are defined via the relations iSzφ Dz(φ)|αi = exp − |αi ¯h ¯h −iφ/2 (Sk)ij ≡ (σk)ij, = e |Sz; ↑ihSz; ↑ |αi 2 iφ/2 +e |Sz; ↓ihSz; ↓ |αi. where the matrix elements are evaluated in the basis In particular: {|Sz; ↑i, |Sz; ↓i}. Dz(2π)|αi = −|αi. For example ¯h S = S = {(|S ; ↑ihS ; ↓ |) + (|S ; ↓ihS ; ↑ |)}, Spin precession 1 x 2 z z z z When the Hamiltonian is so H = ωcSz (S1)11 = (S1)22 = 0 the time evolution operator is ¯h (S1)12 = (S1)21 = , iS ω t 2 U(t, 0) = exp − z c = D (ω t). ¯h z c or ¯h 0 1 (S ) = . Looking at the equations 1 2 1 0 hJxi −→ R hJxi cos φ − hJyi sin φ Thus we get hJ i −→ hJ i cos φ + hJ i sin φ y R y x 0 1 0 −i 1 0 σ = , σ = , σ = . hJzi −→ R hJzi 1 1 0 2 i 0 3 0 −1 one can read that The spin matrices satisfy the anticommutation relations hSxit = hSxit=0 cos ωct − hSyit=0 sin ωct {σi, σj} ≡ σiσj + σjσi = 2δij hSyit = hSyit=0 cos ωct + hSxit=0 sin ωct hSzit = hSzit=0. and the commutation relations We see that [σi, σj] = 2iijkσk. Moreover, we see that Euler’s angles The spinor rotation matrices corresponding to rotations † σi = σi, around z and y axes are det(σ ) = −1, i e−iα/2 0 D (α) 7→ tr(σi) = 0. z 0 eiα/2 cos β/2 − sin β/2 Often the collective vector notation D (β) 7→ . y sin β/2 cos β/2 σ ≡ σ1xˆ + σ2yˆ + σ3zˆ. With the help of Euler’s angles α, β and γ the rotation matrices can be written as is used for spin matrices. For example we get ( 1 ) D(α, β, γ) 7→ D 2 (α, β, γ) = X σ · a ≡ akσk β β e−i(α+γ)/2 cos −e−i(α−γ)/2 sin k 2 2 β β . +a a − ia ei(α−γ)/2 sin ei(α+γ)/2 cos = 3 1 2 . 2 2 a1 + ia2 −a3 We seek for the eigenspinor of the matrix σ · nˆ: and σ · nˆχ = χ. X (σ · a)(σ · b) = σjajσkbk Now j,k sin β cos α X 1 nˆ = sin β sin α , = ({σ , σ } + [σ , σ ]) a b 2 j k j k j k cos β j,k so X −iα = (δjk + ijkiσi)ajbk cos β sin βe σ · nˆ = iα . j,k sin βe − cos β = a · b + iσ · (a × b). The state where the spin is parallel to the unit vector nˆ, is obviously invariant under rotations A special case of the latter formula is −iS·nˆ /h¯ Dnˆ (φ) = e (σ · a)2 = |a|2. and thus an eigenstate of the operator S · nˆ. Now This kind of state can be obtained by rotating the state |Sz; ↑i iS · nˆφ iσ · nˆφ D(nˆ, φ) = exp − 7→ exp − = ¯h 2 1. angle β around y axis, φ φ 2. angle α around z axis, 1 cos − iσ · nˆ sin = 2 2 i.e. φ φ φ cos 2 − inz sin 2 (−inx − ny) sin 2 S · nˆ|S · nˆ; ↑i = S · nˆD(α, β, 0)|Sz; ↑i φ φ φ (−inx + ny) sin 2 cos 2 + inz sin 2 ¯h = D(α, β, 0)|S ; ↑i 2 z and the spinors behave in rotations like ¯h = |S · nˆ; ↑i. iσ · nˆφ 2 χ −→ exp − χ. 2 Correspondingly for spinors the vector β Note the notation σ does not mean that σ would behave cos e−iα/2 ( 1 ) 2 in rotations like a vector, σk −→ Rσk. Instead we have χ = D 2 (α, β, 0)|Sz; ↑i = β iα/2 sin 2 e † X † χ σkχ −→ Rklχ σlχ. is an eigenstate of the matrix σ · nˆ. l For all directions nˆ one has SU(2) As a representation of rotations the 2 × 2-matrices iσ · nˆφ exp − = −1, for any nˆ. ( 1 ) −iσ·nˆ φ/2 D 2 (nˆ, φ) = e 2 φ=2π form obviously a group. These matrices have two characteristic properties: 1. unitarity † −1 ( 1 ) ( 1 ) D 2 = D 2 , 2. unimodularity ( 1 ) 2 D = 1. A unitary unimodular matrix can be written as a b U(a, b) = . −b∗ a∗ The unimodularity condition gives 1 = |U| = |a|2 + |b|2, and we are left with 3 free parameters. The unitarity condition is automatically satisfied because a∗ −b a b U(a, b)†U(a, b) = b∗ a −b∗ a∗ |a|2 + |b|2 0 = = 1. 0 |a|2 + |b|2 Matrices U(a, b) form a group since • the matrix ∗ ∗ U(a1, b1)U(a2, b2) = U(a1a2 − b1b2, a1b2 + a2b1) is unimodular because ∗ ∗ |U(a1a2 − b1b2, a1b2 + a2b1)| = ∗ 2 ∗ 2 |a1a2 − b1b2| + |a1b2 + a2b1| = 1, and thus also unitary. • as a unitary matrix U has the inverse matrix: U −1(a, b) = U †(a, b) = U(a∗, −b). • the unit matrix 1 is unitary and unimodular. The group is called SU(2). Comparing with the previous spinor representation ( 1 ) D 2 (nˆ, φ) = φ φ φ cos 2 − inz sin 2 (−inx − ny) sin 2 φ φ φ (−inx + ny) sin 2 cos 2 + inz sin 2 we see that φ φ Re(a) = cos Im(a) = −n sin 2 z 2 φ φ Re(b) = −n sin Im(b) = −n sin . y 2 x 2 The complex numbers a and b are known as Cayley-Klein’s parameters. Note O(3) and SU(2) are not isomorphic. Example In O(3): 2π- and 4π-rotations 7→ 1 In SU(2): 2π-rotation 7→ −1 and 4π-rotation 7→ 1. The operations U(a, b) and U(−a, −b) in SU(2) correspond to a single matrix of O(3). The map SU(2) 7→ O(3) is thus 2 to 1. The groups are, however, locally isomorphic..