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

2024
Views:

462

Pages:

5
Author:

Jabin F.

Given: Δx = 10^(-15) m ΔpΔx ≈ ħ Calculations: Δp ≈ ħ / Δx Δp ≈ (6.62×10^(-34) Js) / (10^(-15) m) Δp ≈ 6.62×10^(-19) kg m/s p ≈ Δp ≈ 6.62×10^(-19) kg m/s E ≈ p²/2m E ≈ (6.62×10^(-19) kg m/s)² / (2 * 1.67×10^(-27) kg) E ≈ (6.62×10^(-19) kg m/s)² / (3.34×10^(-27) kg) E ≈ 197 MeV Wave Function: ψ(x, 0) = (1 / √(2πσ²)) * exp[-(x²/4σ²) + ik₀x] Short Trick:

≈ ħ / Δx ≈ / (2m) Given: |ψ|^2 = (1 / (2πσ²)) * e^(-x²/2σ²) Calculations: Δx = σ / √2 ΔpΔx = ħ / 2 Δp = ħ / (2σ) (Δp)² = ħ² / (4σ²) = (Δp)² +

² = ħ² / (4σ²) (since

= 0) = / (2m) = ħ² / (8mσ²) Problem: The wave function of a particle of mass m in 3-D position space is given by ψ(r) = (A/r) * e^(ikr), where A and k are constants. Find the corresponding probability current density. Solution: |ψ(r)|² = (A²/r²) * e^(ikr) * e^(-ikr) = A²/r² J = (ħ/2m) * Im(ψ*∇ψ) J = (ħ/2m) * Im(A*e^(-ikr) * ∇(Ae^(ikr))) J = (ħ/2m) * Im(A*e^(-ikr) * (ikA)e^(ikr)) J = (ħ/2m) * Im(-ikA²) J = (ħk/2m) * |A|² Given: ψ(x) = u(x) * e^(ikx) u(x) is real Calculations: J = ψ*∇ψ J = u*(x) * e^(-ikx) * ∇(u(x) * e^(ikx)) J = u*(x) * e^(-ikx) * (u'(x) * e^(ikx) + ik * u(x) * e^(ikx)) J = u*(x) * u'(x) + ik * |u(x)|² ψ*(∂ψ/∂x) = u*(x) * e^(-ikx) * (iku(x) * e^(ikx)) ψ*(∂ψ/∂x) = ik * |u(x)|² Hamiltonian: H = (p²/2m) + V(x) Hψ = (-ħ²/2m) * d²/dx² ψ + V(x)ψ Substituting: Hψ = (-ħ²/2m) * d²/dx² (u(x) * e^(ikx)) + V(x) * u(x) * e^(ikx) Hψ = (-ħ²/2m) * (u''(x) * e^(ikx) + 2ik * u'(x) * e^(ikx) - k² * u(x) * e^(ikx)) + V(x) * u(x) * e^(ikx) Hψ = (-ħ²/2m) * (u''(x) + 2ik * u'(x) - k² * u(x)) * e^(ikx) + V(x) * u(x) * e^(ikx) Hψ = ((-ħ²/2m) * (u''(x) + 2ik * u'(x) - k² * u(x)) + V(x) * u(x)) * e^(ikx) Comparing: E = (-ħ²/2m) * (u''(x) + 2ik * u'(x) - k² * u(x)) + V(x) * u(x) E = (-ħ²/2m) * (u''(x) + 2ik * u'(x)) + (V(x) - (ħ²k²/2m)) * u(x)