2016-03-31 05:44:29 +00:00
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Derivation of Electron States in Hydrogen Atom
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━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━
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2016-03-30 16:23:09 +00:00
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The Hydrogen atom follows the central potential development from the
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previous lecture.
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2016-03-31 05:44:29 +00:00
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❬r,θ,φ❙Ψ❭ = R(r)ₑ,ₗ yₗₘ(θ,φ)
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2016-03-30 16:23:09 +00:00
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2016-03-31 05:44:29 +00:00
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Rₑₗ(r) = Uₑₗ(r) /r
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ρ ≡ √⎛8͟m͟ │E│⎞r
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⎝ ħ² ⎠
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λ = Z͟e͟² √⎛_͟m͟_͟ ⎞
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ħ ⎝2│E│⎠
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2016-03-30 16:23:09 +00:00
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d͟²͟ U - 1͟ l(l+1) + ⎛λ͟ - 1͟⎞U = 0
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dρ² ρ² ⎝ρ 4⎠
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as ρ→0, U→ρˡ⁺¹
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for ρ→∞,
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d͟²͟ U - 1͟U = 0
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dρ² 4
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d͟²͟ U - 1͟U = 0
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dρ² 4
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U(ρ) = A exp(-ρ/2) + B exp(ρ/2) = A exp(-ρ/2)
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(ρ→∞, B=0)
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2016-03-31 05:44:29 +00:00
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U(ρ) = ρˡ⁺¹ exp(-ρ/2) Fₑₗ(ρ)
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2016-03-30 16:23:09 +00:00
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2016-03-31 05:44:29 +00:00
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d͟ U = (l+1)ρˡ exp(-ρ/2) Fₑₗ(ρ)
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2016-03-30 16:23:09 +00:00
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dρ
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2016-03-31 05:44:29 +00:00
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-½ ρˡ⁺¹ exp(-ρ/2) Fₑₗ(ρ)
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+ ρˡ⁺¹ exp(-ρ/2) d͟ Fₑₗ(ρ)
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dρ
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2016-03-30 16:23:09 +00:00
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d͟ U = ⎛l͟+͟1͟ - 1͟ ⎞ U + ρˡ⁺¹ exp(-ρ/2) d͟ F(ρ)
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dρ ⎝ ρ 2 ⎠ dρ
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2016-03-31 05:44:29 +00:00
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d͟²͟ U = -(l͟+͟1͟) U + ⎛l͟+͟1͟ - 1͟⎞² U
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dρ² ρ² ⎝ ρ 2⎠
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+ 2⎛l͟+͟1͟ - 1͟⎞ρˡ⁺¹ exp(-ρ/2) d͟F͟
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⎝ ρ 2⎠ dρ
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+ ρˡ⁺¹ exp(-ρ/2) d͟²͟ F(ρ)
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dρ²
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2016-03-30 16:23:09 +00:00
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2016-03-31 05:44:29 +00:00
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⎡−͟ħ͟² d² + l͟ (l+1)ħ² - Z͟e͟²⎤Uₑₗ(r) = E Uₑₗ(r)
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2016-03-30 16:23:09 +00:00
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⎣2m dr² 2mr² r ⎦
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ρ = √⎛8͟m͟ │E│⎞r
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⎝ ħ² ⎠
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⎛⎞
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⎝⎠
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