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

2024
Views:

153

Pages:

6
Author:

Jabin F.

Wave Function: ψ(x) = { K, (a/2) - (ħ/2m) * d/dx < x < (a/2) + (ħ/2m) 0, otherwise Expectation Values and Uncertainty Δx = √( - ²) = ∫ ψ*(x) x ψ(x) dx / ∫ ψ*(x) ψ(x) dx = ∫ ψ*(x) x² ψ(x) dx / ∫ ψ*(x) ψ(x) dx Calculations = ∫ ψ*(x) x ψ(x) dx / ∫ ψ*(x) ψ(x) dx = 0 (since ψ(x) is even) = ∫ ψ*(x) x² ψ(x) dx / ∫ ψ*(x) ψ(x) dx = ∫ x² |ψ(x)|² dx / ∫ |ψ(x)|² dx = K² * ∫ x² dx / ∫ dx = K² * (x³/3) |₀^d / (x) |₀^d = K² * (d³/3) / d = K² * d²/3 Uncertainty Δx = √( - ²) = √(K² * d²/3 - 0) = K * d / √3 Continuation from the previous image: = (1/3d) * ∫₀^d (a³ + 3a²d + 3ad² + d³) dx = (1/3d) * ((a³ + 3a²d + 3ad² + d³)x) |₀^d = (1/3d) * (a³d + 3a²d² + 3ad³ + d⁴) = (a³ + 3a²d + 3ad² + d²) / 3d = (a + d)² / 3 Uncertainty Product (ΔxΔp) ΔxΔp = √( - ²) * √( -

²) ΔxΔp = √((a + d)² / 3 - 0) * √((ħ²/2m) - 0) ΔxΔp = (a + d) / √3 * √(ħ²/2m) ΔxΔp = (a + d) * ħ / √(6m) Continuation from the previous image: = (-iħ)² * ∫ ψ*(x) d²/dx² ψ(x) dx / ∫ ψ*(x) ψ(x) dx = ħ² * ∫ e^(-x²/2) * d²/dx² (e^(-x²/2)) dx / ∫ e^(-x²/2) * e^(-x²/2) dx = ħ² * ∫ e^(-x²/2) * (-1 + x²) * e^(-x²/2) dx / ∫ e^(-x²/2) * e^(-x²/2) dx = ħ² * ∫ (1 - x²) * e^(-x²) dx / ∫ e^(-x²) dx = ħ² * (∫ e^(-x²) dx - ∫ x²e^(-x²) dx) / ∫ e^(-x²) dx Gaussian Integrals: ∫ e^(-x²) dx = √π ∫ x²e^(-x²) dx = (1/2) * √π Substituting: = ħ² * (√π - (1/2) * √π) / √π = ħ² * (1 - (1/2)) = ħ²/2 Uncertainty Product (ΔpΔx): ΔpΔx = √( -

²) * √( - ²) ΔpΔx = √(ħ²/2 - 0) * √((a + d)² / 3 - 0) ΔpΔx = ħ / √2 * (a + d) / √3 ΔpΔx = ħ(a + d) / √6 Continuation from the previous image: = (1/2) * ∫₀^∞ x²e^(-x²/2) dx / ∫₀^∞ e^(-x²/2) dx Gaussian Integrals: ∫₀^∞ e^(-x²/2) dx = √(π/2) ∫₀^∞ x²e^(-x²/2) dx = √(π/4) Substituting: = (1/2) * (√(π/4) / √(π/2)) = (1/2) * (√(2) / √(4)) = (1/2) * (1 / √(2)) = 1 / (2√(2)) Uncertainty (Δx): Δx = √( - ²) Δx = √((1 / (2√(2))) - 0) Δx = 1 / √(2√(2))