Perturbation Theory Andreas Wacker Mathematical Physics Lund University
General starting point Hamiltonian ^H (t) has typically noanalytic solution of Ψ(t) Decompose Ĥ (t )=Ĥ 0 + V (t) known eigenstates a 0 Ĥ 0 a 0 =E a 0 a 0 hopefully small Eigenstate form basis Ψ(t) = a ψ a (t ) a 0
Stationary perturbation theory, V(t)=V Eigenvalue problem Expand Choose such that is real. Thus Normalization implies
Non-degenerate eigenvalues E⁰ a Admixure of other states Lowering of ground state
Example: Harmonic oscillator with From week 2:
Degenerate level Several states with energy provides Contradictory unless V ji is diagonal! Diagonalize V ji to find correct states Eigenvalues of V ji provide
Procedure 1) Identify orthonormal states i,a ⁰> with Ĥo i,a ⁰ >=E ⁰ a i,a ⁰> which span N-dimensional subspace 2) Evaluate V ji =<j,a ⁰ V i,a ⁰> 3) Determine eigenvalues E n (and -columns c (n) i ) of V ji 4) E 1 na =E n are the energy values in first-order perturbation theory for the states na ⁰>= Σ i c (n) i i,a ⁰>
Example: Stark effect for hydrogen Perturbation V =e ℇ ẑ on second shell with n=2 States 1 = n=2,l=0,m=0, 2 = 2,1,0, 3 = 2,1, 1, 4 = 2,1,1 Calculate 4x4 matrix i V j Fortunately: Calculation provides n=2,1,0 V n=2,0,0 = 3 e ℇa B
Result
Time-dependent perturbation theory V (t)=0 : eigenstates evolve as e i E 0 at /ħ a 0 Determine Transition probability
Interaction picture (Dirac 1927) V=0: Ansatz for finite V: with
Born approximation Ψ(t) =e i Ĥ 0 t /ħ ΨD (t) with i ħ t Ψ D (t) = V D (t) Ψ D (t) Determine P b (t)= b 0 Ψ(t) 2 = b 0 Ψ D (t) 2 for a 0 = Ψ(0) = Ψ D (0) Formal solution Ψ D (t ) = a 0 + 1 t i ħ dt ' V D (t ') Ψ D (t ') 0 = a 0 + 1 t i ħ dt ' V D (t ')[ a0 + 1 t ' 0 i ħ ds V D (s) Ψ D (s) ] 0 Lowest order in V = P b (t )= b 0 Ψ D (t) 2 1 t i ħ dt ' b 0 V D (t ') a 0 2 0
Application: Laser pulse on hydrogen in ground state For 0<t<τ Matrix elements Only possibility for n=2 Integral provides
Result P e (t)= for τ=10π/ω Note P~ ε 0 2 Sharp in ω for long τ
Fermi's golden rule (Dirac 1927) V (t)= F e i ωt provides Math: Define Transition rate Γ a b = P b (t) t with delta-function in energy integral if prefactor is const on energy scale ħ/t = 2 π ħ b0 F a 0 2 δ( E b 0 E a 0 ħ ω)
Fermi's golden rule, continuation V (t )= F e i ωt provides Γ a b = P b (t) = 2 π t ħ b0 ^F a 0 2 δ(e 0 b E 0 a ħω) Constant potential ^V (t)= ^V : Γ a b = 2 π ħ b0 ^V a 0 2 δ(e 0 b E 0 a ) Periodic time-dependence ^V (t)= ^F e i ωt + ^F i ωt e : Γ a b = 2 π ħ b0 ^F a 0 2 δ(e b 0 E a 0 ħω) + 2 π ħ b0 ^ F a 0 2 δ(e b 0 E a 0 +ħ ω) absorption of quanta from the field emission
Example: β-decay of neutron Nuclei, such as n, are eigenstates of strong interaction Ĥ 0 Weak interaction V provides decay into p, e, and ν Γ a b = 2 π ħ b0 ^V a 0 2 δ(e 0 b E 0 a ) Initial state a ⁰>: Final state b ⁰>: P a b = 2π g 2 ħ n with E n =m n c 2 p with E p =m p c 2 (recoil neglected) e with E e = p e 2 c 2 +m e 2 c 4 ν with E ν = ħ 2 k ν 2 c 2 +m ν 2 c 4 Assume (Fermi 1934): b 0 ^V weak interation a 0 2 =g 2 δ [ ħ 2 k ν 2 c 2 +m ν 2 c 4 + p e c 2 +m e 2 c 4 (m n m p )c 2 ]
Evaluating the rate for arbitrary k ν P a pe = d 3 k ν 2 d k ν 4 π k ν 0 Use: b a P a pe = 8π2 g 2 ħ 2 π g 2 ħ δ[ ħ 2 k 2 ν c 2 +m 2 ν c 4 + p e c 2 +m 2 e c 4 (m n m p )c 2 ] = A( p e ) dx g(x)δ [f (x)]= i A 2 ( p e ) m ν 2 c 4 ħ 2 c 2 A( p e ) ħ c A 2 ( p e ) m ν 2 c 4 Θ[ A( p e ) m ν c 2 ] g(x i ) f ' (x i ) where f (x i )=0 and a<x i <b
Example: Radiation transition Elmag. wave: +spin+o(e²) Neglect: kr=o{a B /λ} Dipole d ba = e n' l ' m' r n, l, m =0 unless l ' =l±1, m ' =mor m±1