Variational Method | Santa Fe College

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

2024
Views:

415

Pages:

5
Author:

Jabin F.

Variational Method (Rayleigh-Ritz Method) The variational method is used to Calculate energy of few states (ground" State or some excited states) when Hamiltonian of the system is known but energies & itsteigen fn are unknown. = method for ground states→ <ΨΙΗΙΨ> Choice of Trial wave function:- Include all known information regarding the system (Parity, no. of nodes, behaviour at infinity) in variable part of weave funda & include unknown information through a parameter 'a' known as variation Parameter. or adjustable parameter. Ex: if V(x) = (1/2)mω²x² (known) then at x → ±∞, V → ∞ V(-x) = V(x) → parity definite then choice of Ψ(x) → [e^(-αx²)] (ii) = = <Ψ|H|Ψ> = ∫(ΨHΨ)dx / ∫(ΨΨ)dx = E(α) (iii) minimize → ∂/∂α = 0, ∂²/∂α² > 0 at α = α₀ (iv) = E(α=α₀) → energy & eigen fn of ground state Ψ(x) = Ψ(α₀x) → fn of ground state Quer7 β page-45 V(x) = Kx^β, ψ(x) = e^(-λx) H|Ψ> = EΨ (-ħ²/2m) * d²e^(-λx)/dx² + Kx^β * e^(-λx) = Ee^(-λx) (-ħ²/2m) * (-λ)² * e^(-λx) + Kx^β * e^(-λx) = Ee^(-λx) (-ħ²/2m) * (-λ) * (-λ) * e^(-λx) + Kx^β * e^(-λx) = Ee^(-λx) => (-ħ²/2m) * ((-λ)(-λ)x^(-2) + (-λ)(-λ)x^(-2)) * e^(-λx) + Kx^β * e^(-λx) = Ee^(-λx) => ((ħ²/2m) * λ(λ-1)x^(-2) + λ²x^2 + Kx^β) * e^(-λx) = Ee^(-λx) λ-2 = β => λ = β+2 λ-2 = β => λ = β/2 + 1 (option-a) Ques.8 page-45 E = 9E ħω = 8E = 9E ~ 8E option. (b) = ΣEnPn = (1/4)E + (3/4)E = 8E (option-b) Ques.16 Page-48 =((ℏ)/(2mw)) 1/2 ∗∫[−∞,∞]x 8 ∗e ( −mwx 2 /ħ)dx Ques.21 Page-44 S-line ΔE = hc/λ E = (n²ħ²)/(8ma²) E'n = (n²ħ²)/(2ma²) (λs/λα) = (ΔE/ΔE') => λs = ((ΔE/ΔE') * λα) Ques.14 Page-49 Ψ₀(x) = ((2α)/(π))^(1/4) * e^(-αx²) α = (mw)/(2ħ) = ω²/2 =∫[−∞,∞]∣Ψ 0 (x)∣ 2 ∗x 8 dx = ∫[-∞,∞] |Ψ₀(x)|² * x⁸ dx = ((2α)/(π))^(1/2) * ∫[-∞,∞] e^(-2αx²) * x⁸ dx P.T.O Ques.15 Page-43 P = |<ψ₁'|ψ₁>|^2 <ψ₁'|ψ₁> = sqrt(2/(3L)) * sqrt(2/L) * ∫[0,L] sin(nπx/(3L)) * sin(nπx/L) dx = (2/π) * sqrt(9/2) * (nπ/L) * (2/L)^(1/2) * x^(1/2) * π = (2/π) * a^(1/2) * 2^(1/2) * x^(1/2) * π = (7 * 5 * 3) / (2^8 * a^4) = 105 / (2.56 * a^4) Ques.10 Page-42 |ψ> = ΣCn|φn> = C₀|φ₀> + C₁|φ₁> + C₂|φ₂> <ψ|H|ψ> = (3/2)ħω * (|C₀|^2 + |C₁|^2 + |C₂|^2) = 1 |C₀|^2 + |C₁|^2 + |C₂|^2 = 2/(3ħω) |C₀|^2 + (5/4)|C₁|^2 + (16/5)|C₂|^2 = (11/4)(ħω)^2 |C₀|^2 + (9/4)|C₁|^2 + (25/4)|C₂|^2 = 11 C₀^2 + C₁^2 + C₂^2 = 1