QMexam09eqs852 - Dipole energies: VE = −D · E, VB =...

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Unformatted text preview: Dipole energies: VE = −D · E, VB = −γ S · B, Pauli matrices: S= σz = Rotations: UR (θ ) = e−iJ ·θ , Vector operator: Addition of angular momentum: J = J1 + J2 |j1 j2 m1 m2 = j,mj † UR (θ )V UR (θ ) = R(θ )V D = eR gq γ= 2M σ , σy = 0 −i i0 2 10 0 −1 , σx − 01 10 |j1 j2 jmj j1 j2 jmj |j1 j2 m1 m2 , 2 |j1 j2 jmj → |jmj J 2 |jmj = Two spin 1/2 particles: j (j + 1)|jmj , |j1 − j2 | ≤ j ≤ j1 + j2 S = S1 + S2 , S1z |m1 m2 = m1 |m1 m2 , |m1 m2 = S 2 |sms = 2 sms S2Z |m1 m2 = m2 |m1 , m2 . |sms sms |m1 m2 Sz |sms = ms |ms s(s + 1), Triplet states (symmetric under exchange): |s = 1, ms = 1 = |m1 = 1/2, m2 = 1/2 1 |s = 1, ms = 0 = √ (|m1 = 1/2, m2 = −1/2 + |m1 = −1/2, m2 = 1/2 2 ||s = 1, ms = −1 = |m1 = −1/2, m2 = −1/2 The singlet state (antisymmetric): 1 |s = 0, m = 0 √ (|m1 = 1/2, m2 = −1/2 − |m1 = −1/2, m2 = 1/2 2 Non-degenerate time-independent perturbation theory: (0) H0 |n(0) = En |n(0) , (0) (1) (2) En = En + λEn + λ2 En + . . . , (H0 + λV )|n = En |n |n = |n(0) + λ|n(1) + . . . Vmn − En (0) (0) (1) En = n(0) |V |n(0) |n(1) = − m=n |m(0) (0) Em (2) En = − m=n Em − En (0) |Vmn |2 Degenerate time-independent perturbation theory: (0) H0 |nm(0) = En |nm(0) , (0) (1) (2) Enm = En + λEnm + λ2 Enm + . . . , (H0 + λV )|nm = Enm |nm |nm = |nm(0) + λ|nm(1) + . . . dn′ (2) Enm (In V In − Atomic physics: AC Stark: (1) Enm )|nm(0) = 0, = n′ =n m′ =1 En′ − En |Vn′ m′ nm |2 (0) (0) V = −D · E Zeeman: V = −γ S · B Fine-structure: Interaction type: VSO = V (R)L · S J =L+S Good basis: |nℓsjmj Degeneracy of j -states lifted at first-order: ESO = nℓ|V (R)|nℓ ⊗ Exact fine-structure result: (0) Enℓj = |En | (1) (1) 1 3 j (j + 1) − ℓ(ℓ + 1) − 2 4 2n 3 − j + 1/2 2 α2 2n4 Hyperfine structure: Interaction type: Vhf = V (R)I · J Good basis: |nℓsijf mf Wigner Eckert theorem: for any vector operator V , kjm|Vz |k′ j ′ m′ = 0 unless m = m′ kjm|V± |k′ j ′ m′ = 0 unless j = j ′ Vkj = Ikj V Ikj Vkj = a(k, j )Jjk a(k, j ) = kj |J · V |kj 2 j (j + 1) Zeeman effect: VB = ω0 (Lz + 2Sz ) where ω0 = |e|B0 /(2me ) is the Larmor frequency Strong-field Zeeman effect: ω0 ≫ Ef ine good basis: |nℓsmℓ ms EB = ω0 (mℓ + 2ms ), Intermediate Zeeman effect ω0 ∼ Ef ine good basis: |nℓsjm EB = ω0 gj mj g-factor (Wigner Eckert theorem applied to VB in j subspace): gj = 3 s(s + 1) − ℓ(ℓ + 1) J · (L + 2S ) =+ 2 j (j + 1) 2 2j (j + 1) Weak-field Zeeman effet ω ≪ Ef ine good basis: |nℓsijf mf EB = ω gf mf g-factor (Wigner Eckert theorem applies to VB in f subspace: gf = F · (L + 2S ) = gj 2 f (f + 1) 2 f (f F ·J 1 j (j + 1) − i(i + 1) = gj 1 + + 1) 2 f (f + 1) Time-Dependent Perturbation theory: Interaction picture: U0 (t) = e−iH0 t/ † |ψI (t) = U0 (t)|ψS (t) † OI (t) = U0 (t)OS U0 (t) d i UI (t) = − VI (t)UI (t) dt UI (t) = UI (t) + λUI (t) + . . . UI (t) = (j ) (0) (1) j 0 −i t t1 tj −1 dt1 0 dt2 . . . 0 dtj VI (t1 )VI (t2 ) . . . VI (tj ) US (t) = U0 (t)UI (t), US (t) = U0 (t) + λ = + λ2 −i −i 0 2 0 t t |ψS (t) = US (t)|ψS (0)|ra dt1 U0 (t − t1 )V (t1 )U0 (t1 ) t2 0 dt2 dt − 1U0 (t − t − 2)V (t2 )U0 (t2 − t1 )V (t1 )U0 (t1 ) + . . . Transition amplitudes: cnm (t) = n|US (t)|m = e−iωn t n|UI (t)|m c(0) = e−iωn t δn,m nm i 0 t i c(1) = − e−iωn t nm t 0 dt1 n|VI (t)|m = − Pm→n (t) = λ 2 dt1 e−iωn (t−t1 )t n|V (t)|m e−iωm t1 n=m |c(1) (t)|2 nm |c(1) (t)|2 , nm n=m Pm→m (t) = 1 − λ2 Fermi Golden rule: Γ= 2π |Gif |2 n(Ei ) ...
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This note was uploaded on 10/25/2010 for the course PHYSICS PHYS 851 taught by Professor Michaelmoore during the Fall '08 term at Michigan State University.

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