Angular Momentum. Andreas Wacker Mathematical Physics Lund University
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1 Angular Momentum Andreas Wacker Mathematical Physics Lund University
2 Commutation relations of (orbital) angular momentum Angular momentum in analogy with classical case L= r p satisfies commutation relations [ L x, L y ]=i ħ L z [ L y, L z ]=i ħ L x [ L z, L x ]=i ħ L y and consequently [ L 2, L i ]=0 for i=x, y, z Is there a deeper meaning behind?
3 Relation to rotations in 3 dimensions Rotation of position Rotation of wavefunction Finite angle General operator for rotations in 3 dim space
4 Does the sequence of rotations matter? Û x / y rotation around x/y-axis with angle φ x / y Commutator [ L x, L y ] 0 implies The order of rotations matters!
5 Symmetry of system manifests in commutation relations Symmetry around x-axis Rotated states Φ i =Û x Ψ i Matrix elements Ψ 1 Ĥ Ψ 2 and Φ 1 Ĥ Φ 2 are equal Û x Ĥ Û x =Ĥ As Û x is unitary Ĥ Û x =Û x Ĥ The same holds for small angles: Invariance under rotation around x-axis implies Commutator [ Ĥ, L x ]=0
6 Generalized angular momentum Common eigenstates Hermitian operators Ĵ x, Ĵ y, Ĵ z satisfying [Ĵ j, Ĵ k ]=i ħ l ϵ jkl Ĵ j Ĵ 2 =Ĵ x 2 + Ĵ y 2 + Ĵ z 2 satisfies [Ĵ 2, Ĵ j ]=0 Eigenvalues of Ĵ 2 are not negative Search set of common eigenstates j, m satisfying Ĵ 2 j, m = j ( j +1)ħ 2 j,m and Ĵ z j,m =m ħ j,m with j 0
7 Properties of Ψ + =Ĵ + j, m Ĵ 2 j, m = j ( j +1)ħ 2 j,m and Ĵ z j, m =m ħ j,m with j 0 Define Ĵ + =Ĵ x +i Ĵ y and Ĵ - =Ĵ x i Ĵ y =Ĵ + Ψ + Ψ + =[ j ( j+1) m(m+1)]ħ 2 (a) m j (b) Ψ + = null m= j or m= j 1 (c) Ĵ 2 Ψ + = j ( j+1)ħ 2 Ψ + and Ĵ z Ψ + =(m+1)ħ Ψ + Identify j, m+1 = 1 ħ j ( j+1) m(m+1) Ĵ + j, m Chain stops only if (j-m) is a natural number
8 Properties of Ψ - =Ĵ - j, m Ĵ 2 j, m = j ( j +1)ħ 2 j,m and Ĵ z j,m =m ħ j,m with j 0 Define Ĵ + =Ĵ x +i Ĵ y and Ĵ - =Ĵ x i Ĵ y =Ĵ + Ψ - Ψ - =[ j ( j+1) m(m 1)]ħ 2 (a) m j (b) Ψ - = null m= j or m= j+1 (c) Ĵ 2 Ψ - = j ( j +1)ħ 2 Ψ - and Ĵ z Ψ - =(m 1)ħ Ψ - Identify j,m 1 = 1 ħ j ( j +1) m(m 1) Ĵ - j, m Chain stops only if (j+m) is a natural number
9 Summary: Common eigenstates of Ĵ 2 and Ĵ z j-m, j+m N 0 2j N 0 Sequence of states j,m>, m=-j,-j+1,...,j 1 j, m+1 = ħ j ( j+1) m(m+1) Ĵ + j,m j, m 1 = 1 ħ j ( j+1) m(m 1) Ĵ - j,m Ĵ 2 j, m = j ( j +1)ħ 2 j,m and Ĵ z j,m =m ħ j,m with j=0, 1 2,1, 3 2, and m= j, j+1, j form multiplets M j with 2j+1 states
10 Eigenstates of Hamiltonian with full rotational symmetry Rotational symmetry: [Ĥ,Ĵ i ]=0 for i=x,y,z and thus [Ĥ,Ĵ 2 ]=0 There exists a common set of eigenstates of Ĥ,Ĵ z, and Ĵ 2 All elements of the multiplet j,m>, m=-j,-j+1,...j have the same energy (2j+1)-fold degeneracy
11 Example: Matrix representation of M 1 1 j, m+1 = ħ j ( j+1) m(m+1) Ĵ + j,m 1 j, m 1 = ħ j ( j+1) m(m 1) Ĵ - j,m
12 Orbital angular momentum spherical harmonics Spatial representation of ^rxp ^ in spherical coordinates m and consequently also j l must be integer! Normalization For example
13 Spin: two-fold degeneracy implies j=1/2 j, m+1 = j, m 1 = 1 ħ j ( j+1) m(m+1) Ĵ + j,m 1 ħ j ( j+1) m(m 1) Ĵ - j,m Notation: Ĵ i Ŝ i
14 Magnetic moment is related to angular momentum of a particle classical homogeneously charged particle μ= q 2m L Define gyromagnetic ratio (Landé-factor, g-factor) atomic physics dominated by electrons μ= g e Ŝ with e>0 2m e Pure electron spin: g e = (quantum electrodynamics) nuclear physics dominated by protons μ=g Proton: g p = Neutron: g n = e 2m p Ŝ
15 Dynamics in a magnetic field Spin 1 2 : Ŝ ħ 2 [( ) e x + ( 0 i i 0 ) e y + ( ) e z] Precession with Larmor frequency ω L =g e eb/2m e
16 Coupling between two spins: The product space Positronium: Bound system of one electron and one positron Arbitrary state Ŝ ħ 2 [( ) e x + ( 0 i i 0 ) e y + ( ) e z] a i Ĥ a j = U 4 ( ) Ground state: E=-3U/4, J=0, para-positronium τ=125ps E=U/4, J=1, ortho-positronium τ=142ns (U=0.8 mev)
17 Spin-Orbit coupling Product space of space and spin spanned by Spinor representation Search for eigenstates classified by angular momentum
18 System 1: Common eigenstates of L 2, L z, Ŝ 2, Ŝ z Possible as [Ŝ i, L j ]=0 Denote states as a,l, m l, s,m s 1 Example
19 The total angular momentum Ĵ= L+Ŝ Ĵ z = L z +Ŝ z Ĵ 2 = L 2 +Ŝ 2 +2 L z Ŝ z + L + Ŝ - + L - Ŝ + Calculation gives System 1 can not be chosen as eigenstates of Ĵ 2
20 System 2: Common eigenstates of Ĵ2, Ĵ z, L 2, Ŝ 2 Ĵ z = L z +Ŝ z Ĵ 2 = L 2 +Ŝ 2 +2 L z Ŝ z + L + Ŝ - + L - Ŝ + Ĵ 2, Ĵ z, L 2, Ŝ 2 all commute with each other Denote states as a, j, m j,l, s 2 Change of basis Clebsch-Gordan coefficients
21 Example: Determine the Clebsch-Gordan coefficients of Ĵ z = L z +Ŝ z Ĵ 2 = L 2 +Ŝ 2 +2 L z Ŝ z + L + Ŝ - + L - Ŝ + Ĵ + j, m =ħ j ( j+1) m(m+1) j, m+1 Ĵ - j, m =ħ j ( j+1) m(m 1) j, m 1 In spinor representation
22 Fine-structure of the H-atom Spin-orbit interaction: Ĥ SO = f (r) L Ŝ= 1 2 f (r) (Ĵ 2 L 2 Ŝ 2 ) [ Ĥ SO, L z ] 0 and [ Ĥ SO, Ŝ z ] 0 System 1 is not a good basis [ Ĥ SO, Ĵ 2 ]=0, [ Ĥ SO, Ĵ z ]=0, [ Ĥ SO, L 2 ]=0, [ Ĥ SO, Ŝ 2 ]=0 System 2 is a good basis Ground state: n=1,l=0,s=1/2 implies j=1/2, no effect of Ĥ SO Excited state: n=2,l=1,s=1/2 allows j=1/2, j=3/2 2p 1/2 is 45μeV lower than 2p 3/2
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