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

2024
Views:

212

Pages:

7
Author:

Jabin F.

# Multi dimension potential in Cartesian co-ordinates:==> 3-D V(r) = V(x,y,z) Schrodinger eqn. 2 ΗΨ(x,y,z) = ΕΨ(x,y,z) -²+ √(x,y,z)) (x,y,z) = EY(x,y,z) 2m 2 2m ²+ ち2 = 0 Separation variable method ↓ IF V(x, y, z) = V(x) + V(y) + V(z) Ψ(x, y, z) = Ψ(x)Ψ(y)Ψ(z) -ħ²/2m (∂²/∂x² + ∂²/∂y² + ∂²/∂z²) Ψ(x)Ψ(y)Ψ(z) + (V(x) + V(y) + V(z)) Ψ(x)Ψ(y)Ψ(z) = E Ψ(x)Ψ(y)Ψ(z) ⇒ -ħ²/2m (Ψ(x)Ψ(y)Ψ(z)) ∂²/∂x² (Ψ(x)) + Ψ(x)Ψ(y)Ψ(z) ∂²/∂y² (Ψ(y)) + Ψ(x)Ψ(y)Ψ(z) ∂²/∂z² (Ψ(z)) + (V(x) + V(y) + V(z)) Ψ(x)Ψ(y)Ψ(z) = E Ψ(x)Ψ(y)Ψ(z) devide by Ψ(x)Ψ(y)Ψ(z) -ħ²/2m (1/Ψ(x) ∂²/∂x² (Ψ(x)) + 1/Ψ(y) ∂²/∂y² (Ψ(y)) + 1/Ψ(z) ∂²/∂z² (Ψ(z))) + V(x) + V(y) + V(z) = E ∂²Ψ(y) ------- + 2m/ħ² (E_y - V(y)) Ψ(y) = 0 ∂y² ∂²Ψ(z) ------- + 2m/ħ² (E_z - V(z)) Ψ(z) = 0 ∂z² IF V(x, y, z) = V(x) + V(y) + V(z) Ψ(x, y, z) = Ψ(x)Ψ(y)Ψ(z) E = Ex + Ey + Ez Probability |Ψ|^2 = |Ψ(x)|^2 |Ψ(y)|^2 |Ψ(z)|^2 IF V(x, y, z) ≠ V(x) + V(y) + V(z) Ψ(x, y, z) ≠ Ψ(x)Ψ(y)Ψ(z) E ≠ Ex + Ey + Ez @web18 page48 ψ(x) = (1/π)^(1/4) * (1 + √2*x) * exp(-x^2/2) ∴ ψn(x) = (α/πn!)^(1/2) * e^(-α^2x^2/2) * Hn(αx) ψ(0) = (α^2/π)^(1/4) * e^(-α^2x^2/2) ψ'(0) = (α^2/4π)^(1/4) * √2 * α * x * e^(-α^2x^2/2) = (α^2/π)^(1/4) * √2 * α * x * e^(-α^2x^2/2) α² = mω = 1/x² Ψ(η) = (1/πx²)^(1/4) e^(-η²/2x²) + (1/πx²)^(1/4) (1/x) η e^(-η²/2x²) Ψ(η) = Ψ₀(η) + Ψ₁(η) |Ψ> = 1|Φ₀> + 1|Φ₁> = <Ψ|x|Ψ> / <Ψ|Ψ> = 1/2 (<Φ₀|x|Φ₀> + <Φ₁|x|Φ₁> + <Φ₀|x|Φ₁> + <Φ₁|x|Φ₀>) = 1/2 (0 + 0 + √ħ/2mw + √ħ/2mw) = √ħ/2mw = μ/2 for l = 100 μm = 100 / √2 = 0.71 μm Ans. @yes 19 page-48 ψ(x,t) = -ħ²/2m ∫ψ*(x,t) (d²/dx²) ψ(x,t) dx = T ∫ψ*(x,t) ψ(x,t) dx (i) = ∫ψ*(x,t) (1/2) mw²x² ψ(x,t) dx = V ∫ψ*(x,t) ψ(x,t) dx (ii) for state ψ(n,t) v(n,1) = 1/2 mw²n² For Ψ(λm, t) = ∫(Ψ*(λm,t) / (2m) * (d²/dλm²) Ψ(λm,t)) dλm -∞ ∫Ψ*(λm, t) Ψ(λm, t) dλm -∞ λ'm = λm'' ⇒ m = λ'm / λ dλm = (1/λ) dλm' d/dλm = λ d/dλm' d²/dλm² = d/dλm (d/dλm) = λ² d²/dλm'² = λ² ∫(ψ*(λm',t) (-ħ²/2m) (d²/dλm'²) ψ(λm',t)) dλm' -∞ ∫ ψ*(λm',t) ψ(λm',t) dλm' -∞ = λ² ∫ ψ*(λm',t) (-ħ²/2m) (d²/dλm'²) ψ(λm',t) dλm' -∞ ∫ ψ*(λm',t) ψ(λm',t) dλm' -∞ from (i) and (iii) = λ²T NOW, = ∫(ψ*(λm, t) * (1/2) * m * w² * λm² * ψ(λm, t)) dλm -∞ ---------------------------------------- ∫ ψ*(λm, t) * ψ(λm, t) dλm -∞ Put λm = λm'/λ Similarly = (1/λ²) ∫(ψ*(λm', t) * (1/2) * m * w² * λm'² * ψ(λm', t)) dλm' -∞ ----------------------------------------------------- ∫ ψ*(λm', t) * ψ(λm', t) dλm' -∞ from (i) and (iv) = V/λ² Option (A)