lec+12+2013+158a+complete

lec+12+2013+158a+complete - F Grieman Hydrogen-like atoms...

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Hydrogen-like atoms Angular Wavefunctions Solving Radial Schroedinger Equation One Electron Wave Functions Exam Friday!!! Closed Book Bring Pencils, No Calculator Likely 55 minutes, Come by 9:55 a.m. Q&A Wednesday, Time?? F. Grieman 1/10
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Last time: H atom Wave Function: Ψ(r,θ, ) = R(r) Θ(θ) Φ( ) = R(r)Y(θ, ) Separate Sch. Eq. – angular part just what we had before L 2 = ℓ(ℓ+1)ħ 2 ; ℓ = 0, 1, 2, … L z = mħ ; m = 0, ±1, ±2, … ±ℓ Y ℓ,m (θ, ) = Θ ℓ, m (θ) Φ m ( ) ; Φ m ( ) = (2π) -1/2 e im Using L + we found Θ ℓ,ℓ = N sin θ; Y ℓ,ℓ = N ℓ,ℓ sin θ (2π) -1/2 e iℓ Using L - we can find the rest For example: Y 1,1 = (3/8π) 1/2 sin θ (2π) -1/2 e i Y 1,0 = (3/4π) 1/2 cos θ Y 1,-1 = (3/8π) 1/2 sin θ (2π) -1/2 e -i Simple math functions – (called Spherical Harmonics) Let’s look at them 2/10
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3/10
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4/10 p -1
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Separation of variables via product function: ψ(r,θ, ) = R(r) Y(θ, ) Last time Other side of equation Radial Kinetic Angular Kinetic Potential Energy Energy Energy
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