University:
Santa Fe College
Course:
PHY 2048 | General Physics 1 with Calculus
Academic year:

2024
Views:

487

Pages:

7
Author:

Jabin F.

Time Independent Perturbation Theory --> ① Non-deg. Time independent pert. Theory --> Ĥ = Ĥ₀ + Ĥₚ Ĥ₀ = Unperturbed Hamiltonian (S.E is exactly solvable) Extracted Text: Ĥ = Perturbed Hamiltonian. = (S.E is not solvable) Ĥ << Ĥ₀ or Ĥ *cond" of applicability Ĥₚ = λĤ' λ = Unitless parameter << 1 let |φₙ> be the energy eigen fn of unperturbed Hamiltonian Ĥ₀ with eigen energy Eₙ^(0) Ĥ₀|φₙ> = Eₙ^(0)|φₙ> let |ψₙ> be the eigen fn of total Hamiltonian with energy Eₙ Ĥ|ψₙ> = Eₙ|ψₙ> |ψₙ> - Unknown. Eₙ = Eₙ^(0) + λEₙ^(1) + λ²Eₙ^(2) + ... |ψₙ> = |φₙ> + λ|ψₙ^(1)> + λ²|ψₙ^(2)> + ... E(K)m → K-th order Correction fm energy |ψₙ(K)> → K-th order Correction fm wavefn of n-th state. as Ĥₚ << Ĥ₀ or Ĥ, |φₙ> and |ψₙ> are normalised. <φₙ|φₙ> = 1 <ψₙ|ψₙ> = 1 |ψₙ> is also normalised. => <ψₙ|ψₙ> = <φₙ|φₙ> + λ(<φₙ|ψₙ^(1)> + <ψₙ^(1)|φₙ>) λ²( <φₙ|ψₙ^(2)> + <ψₙ^(2)|φₙ> + <ψₙ^(1)|ψₙ^(1)> ) λ³( <φₙ|ψₙ^(3)> + <ψₙ^(3)|φₙ> + <ψₙ^(1)|ψₙ^(2)> + <ψₙ^(2)|ψₙ^(1)> ) ... 1 = 1 + λ( ... ) + λ²( ... ) + λ³( ... ) + ... => <φₙ|ψₙ^(1)> = 0 = <ψₙ^(1)|φₙ> <φₙ|ψₙ^(2)> = 0 = <ψₙ^(2)|φₙ> <φₙ|ψₙ^(3)> = 0 = <ψₙ^(3)|φₙ> Ĥ|ψₙ> = Eₙ|ψₙ> Put value ψₙ (Ĥ₀ + Ĥₚ)(|φₙ> + λ|ψₙ^(1)> + λ²|ψₙ^(2)> + ...) = Eₙ(|φₙ> + λ|ψₙ^(1)> + λ²|ψₙ^(2)> + ...) (coeff of λ⁰) Ĥ₀|φₙ> = Eₙ^(0)|φₙ> (coeff of λ¹) Ĥ₀|ψₙ^(1)> + Ĥₚ|φₙ> = Eₙ^(0)|ψₙ^(1)> + Eₙ^(1)|φₙ> (coeff of λ²) Ĥ₀|ψₙ^(2)> + Ĥₚ|ψₙ^(1)> = Eₙ^(0)|ψₙ^(2)> + Eₙ^(1)|ψₙ^(1)> + Eₙ^(2)|φₙ> First order correction in energy of nth state: E_n^(1) = <φ_n|H'|φ_n> / <φ_n|φ_n> The energy of n-th state corrected to first order: E_n = E_n^(0) + λE_n^(1) E_n = E_n^(0) + <φ_n|H'|φ_n> / <φ_n|φ_n> E_n^(1) = <φ_n|H'|φ_n> / <φ_n|φ_n> is the expectation value of H' over the unperturbed state |φ_n>. First order correction in wave function: |ψ_n^(1)> = Σ_(m≠n) <φ_m|H'|φ_n> / (E_n^(0) - E_m^(0)) |φ_m> The wave function corrected to first order: |ψ_n> = |φ_n> + λ|ψ_n^(1)> |ψ_n> = |φ_n> + Σ_(m≠n) <φ_m|H'|φ_n> / (E_n^(0) - E_m^(0)) |φ_m> The Second order Correction in energy → E_n'' = Σ_(m≠n) |<φ_m|H'|φ_n>|^2 / (E_n' - E_m')^2 ground state The Second order Correction in energy is either zero or -ve. The energy (corrected to second order)→ E_n = E_n^(0) + λE_n^(1) + λ²E_n^(2) E_n = E_n^(0) + <φ_n|H'|φ_n> + Σ_(m≠n) |<φ_m|H'|φ_n>|^2 / (E_n^(0) - E_m^(0))^2 Prob. Consider a particle of mass 'm' that is moving in 1-D potential V(x) = { 0, 0 ψ(x) = √(2/a) sin(nπx/a) (ii) H' = { V₀, 0 φ_n^(1)(x) = { √(2/a) * sin(nπx/a), if 0 < x < a ; 0, otherwise } ① En^(1) = <φ_n|Hp|φ_n> = V0 * ∫(0 to a/2) (2/a) * sin^2(nπx/a) dx = V0/2 The energy corrected to first order En = En^(0) + V0 En = (n^2 * pi^2 * hbar^2) / (2ma^2) + V0/2 ② En^(1) = <φ_n|Hp|φ_n> = V0 * ∫(0 to a/2) (2/a) * sin^2(nπx/a) dx + V0 * ∫(a/2 to a) (2/a) * sin^2(nπx/a) dx = V0 The energy corrected to first order En = En^(0) + V0 En = (n^2 * pi^2 * hbar^2) / (2ma^2) + V0