lsho - Numerical Solution 1. Start at x = −xmax . •...

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Unformatted text preview: Numerical Solution 1. Start at x = −xmax . • Specify the wave function and derivative: Ψ(x) = 10−6 Ψ (x) ≡ • Choose an energy arbitrarily E = 0.3 2. Integrate forward with ∆x Ψ(x + ∆x) Ψ (x + ∆ x) dΨ (x) dx dΨ = 10−6 dx o = 0.01 until xmax Ψ(x) + Ψ (x) ∆x dΨ (x) Ψ (x) + ∆x dx (1) (2) ¯ = −2(E − v (x))Ψ(x) ⇐ The Schrodinger equation (3) Wave Function Ψ(x) 8 7 6 5 4 3 2 1 0 Ψ(x) V(x) / εo Potential 0.30 εo -4 -3 -2 -1 0 X/L 1 2 3 4 -4 -3 -2 -1 0 X/L 1 2 3 4 Numerical Solution 1. Choose energy E (a) Start at left end. (b) Integrate forward to right end. 2. Change the Energy and repeat step 1 (a) For most energies: Ψ(x) → ±∞ as x → ∞. This is unphysical. Want to find the discrete E where: Ψ(x) → 0 as x → +∞ Wave Function Ψ(x) 8 7 6 5 4 3 2 1 0 4.70 ε 4.65 εo 4.60 4.55 4.50 εo 4.45 4.40 4.35 4.30 εo 4.25 4.20 4.15 εo 4.10 4.05 4.00 εo 3.95 3.90 3.85 3.80 εo 3.75 3.70 3.65 εo 3.60 3.55 3.50 εo 3.45 3.40 3.35 3.30 εo 3.25 3.20 3.15 εo 3.10 3.05 3.00 εo 2.95 2.90 2.85 2.80 εo 2.75 2.70 2.65 εo 2.60 2.55 2.50 εo 2.45 2.40 2.35 2.30 εo 2.25 2.20 2.15 εo 2.10 2.05 2.00 εo 1.95 1.90 1.85 1.80 εo 1.75 1.70 1.65 εo 1.60 1.55 1.50 εo 1.45 1.40 1.35 1.30 εo 1.25 1.20 1.15 εo 1.10 1.05 1.00 εo 0.95 0.90 0.85 0.80 εo 0.75 0.70 0.65 εo 0.60 0.55 0.50 εo 0.45 0.40 0.35 0.30 o -4 -3 -2 -1 0 1 Ψ(x) V(x) / εo Potential 2 3 4 -4 -3 -2 -1 X/L 0 1 2 X/L Find physical wave functions when 1 3 5 En = o , o , o . . . 2 2 2 1 = n+ n = 0, 1, 2, 3 . . . o 2 o ≡ ¯ ωo h 3 4 ...
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