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

2024
Views:

257

Pages:

7
Author:

Jabin F.

Probability density A = B P_I = |Ψ_I|^2 = 4|A|^2 * cos^2(k_I * x) P_II = |Ψ_II|^2 = |C|^2 2x = 0 Prob. - V(x) = { 0, 0 < x < a ; ∞, otherwise } for ground state ψ₁(x) = { √(2/a) * sin(πx/a), 0 < x < a ; 0, otherwise } P = ∫ ψ₁*(x) ψ₁(x) dx (i) P = ∫₀^(a/2) ψ₁*(x) ψ₁(x) dx = (2/a) ∫₀^(a/2) sin²(πx/a) dx = (2/a) ∫₀^(a/2) (1 - cos(2πx/a)) / 2 dx = (1/a) [x - sin(2πx/a) / (2π/a)]₀^(a/2) = 1/2 |\ψ₁|^2 P a 0 a/2 (ii) p = integral from a/2 to a of |ψ₁|^2 dx = 1/2 (iii) p = integral from 0 to a/4 of |ψ₁|^2 dx = (2/a) integral from 0 to a/4 of sin^2(πx/a) dx = (2/a) integral from 0 to a/4 of (1 - cos(2πx/a))/2 dx = (1/a) [x - sin(2πx/a)/(2π/a)] from 0 to a/4 = (1/a) [a/4 - (a/2π)(1-0)] = 1/4 - 1/(2π) = integral from 0 to a/4 of |ψ₁|^2 dx p = 1/4 - 1/(2π) (iv) p = integral from a/4 to a/2 of |ψ₁|^2 dx = (2/a) integral from a/4 to a/2 of (1 - cos(2πx/a))/2 dx p = 1/4 + 1/(2π) = 0.81 H.U.P For ∆x. ∆p For n-th state, ∆x. ∆p = ? ψₙ(x) = √(2/a) * sin(nπx/a), 0 <= x <= a 0, otherwise = integral from -∞ to ∞ of |ψₙ(x)|² x dx = integral from 0 to a of (2/a) x * sin²(nπx/a) dx = integral from -∞ to ∞ of |ψₙ(x)|² x² dx = integral from 0 to a of (2/a) x² * sin²(nπx/a) dx = (2/a) ∫₀ᵃ x(1 - cos(2nπx/a)) dx = (1/a) (∫₀ᵃ x dx - ∫₀ᵃ x cos(2nπx/a) dx) = (1/a) (a²/2 - 2x sin(2nπx/a) / (2nπ/a) + cos(2nπx/a) / (2nπ/a)² )₀ᵃ = a/2

= (2/a) ∫₀ᵃ sin(nπx/a) (ih/dx) sin(nπx/a) dx

= 0 ψₙ(x) = √(2/a) sin(nπx/a) Eigenfunction of linear momentum operator

= Σₙ pₙλₙ = (i/2) (nπħ/a) + (i/2) (-nπħ/a)

= 0 = (2/a) ∫₀ᵃ x sin²(nπx/a) dx = (2/a) x (1/2) x ∫₀ᵃ (1 - cos(2nπx/a)) dx = a²/3 - a²/(n²π²) = (2/a) ∫₀ᵃ sin(nπx/a) (-ħ²/dx²) sin(nπx/a) dx = +ħ² (2/a) (n²π²/a²) ∫₀ᵃ sin²(nπx/a) dx = (n²π²ħ²/a²) x (2/a) ∫₀ᵃ sin²(nπx/a) dx = n²π²ħ²/a² => = 2m K.E = p²/2m = n²π²ħ²/2ma² = Eₙ = 2m for n-th state = n²π²ħ²/a² Now, δx = √( - ²) δx = a√(1/12 - 1/2n²π²) ΔP = √( -

²) ΔP = nπħ/a Δx.ΔP = nπħ√(1/12 - 1/2n²π²) Δx.ΔP = 0.568ħ for n = 1