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Momentum eigenstates å å å å linear momentum eigenstates orbital angular momentum eigenstates spin angular momentum eigenstates addition of angular momenta l the generic angular momentum operator We will work mainly with quantum numbers in the applications RAF211 - CZJ 1 Linear momentum eigenstates We will consider motion in one dimension (x): h d p x = – i -----2π d x By solving the eigenvalue equation p x ψ = p x ψ (where both ψ and p x are unknown), we find ψ( x) = c ⋅ e RAF211 - CZJ 2πi -------- xp x h 2 Eigenstates of orbital angular momentum We will look for the simultaneous eigenstates ψ 2 of the operators L and L z 2 2 Lz ψ = Lz ψ , L ψ = L ψ z We will use spherical coordinates: (r,θ,φ) x RAF211 - CZJ θ r y φ 3 Lz eigenstates and eigenvalues The eigenvalue equation h dψ – i -----= L z gives 2π d φ ψ ( φ ) = ce 2πi -------- φL z h quantized h Condition: ψ ( φ ) = ψ ( φ + 2π ) ⇒ L z = m -----2π m = 0, ± 1, ± 2, … is the magnetic quantum number RAF211 - CZJ 4 Eigenstates of L2 and Lz Spherical harmonics m ( 2l + 1 ) ( l – m )! 1 ⁄ 2 m imφ Y lm ( θ, φ ) = ( – 1 ) -------------------------------------P l ( cos θ )e 4π ( l + m )! m where P l ( cos θ ) are the associated Legendre functions 2 h 2 L = ------ l ( l + 1 ) , l = 0, 1, 2, … the orbital quantum 2π number the magnetic quantum number is restricted by l : m = – l ,– l + 1 ,…, 0, … ,l – 1 ,l RAF211 - CZJ 5 Parity eigenstates The spherical harmonics are also eigenstates of the parity operator l PY lm = ( – 1 ) Y lm Application: transitions in atoms RAF211 - CZJ 6 The generic angular momentum operator h- J and cyclic permutations J = J x u x + J y u y + J z u z , [ J x, J y ] = i ----2π z 2 h h- , j = 0, 1, 2, … , m = – j ,– j + 1 ,… ,j J = ------ j ( j + 1 ) , L z = m ---- 2π j 2π 2 By solving the eigenvalue equations for J and J z simultaneously and assuming n eigenstates, we find that there are two ‘types’ of angular momentum: a integer j a half-integer j 2 RAF211 - CZJ 7 The spin angular momentum Spin does not involve a rotation in space 2 h S = ------ s ( s + 1 ) 2π 2 where s is the spin quantum number h S z = m s ------ , 2π m s = – s ,– s + 1 ,… ,s – 1 ,s a integer s : bosons a half-integer s : fermions RAF211 - CZJ 8 Addition of angular momenta Let us assume that we want to ‘add’ the quantum numbers l and s to find the quantum number of the total angular momentum, j , and the quantum number of the z-component, m j The rules are: 1. j = l – s , …, l + s 2. for each j , we have 2j + 1 m j ’s: m j = – j ,– j + 1 ,… ,j Application: L-S coupling and J-J coupling in atoms RAF211 - CZJ 9