# l15 - The atom Spherical coordinates Radial wavefunctions n...

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Unformatted text preview: The atom Spherical coordinates Radial wavefunctions n and ℓ Spherical harmonics Selection rules Atom wavefunctions Angular momentum Filling states The atom I Let’s now tackle an especially relevant problem in quantum mechanics: the solution of electron orbitals! (Chapter 8 of Serway). • We expect to have some constants of motion: 1 Total kinetic energy: quantum number is n 2 Total angular momentum L : quantum number is ℓ 3 Projection of L onto one axis: quantum number is m ℓ • The time-independent Schrödinger equation in multiple dimensions involves a Laplacian ∇ 2 : − planckover2pi1 2 2 m ∇ 2 ψ + U ( r ) ψ = E ψ • We’ll use spherical coordinates ( r , θ, φ ) . The Laplacian or second derivative ∇ 2 in spherical coordinates ( r , θ, φ ) becomes (equivalent but different form from Serway) ∇ 2 = 1 r 2 ∂ ∂ r ( r 2 ∂ ∂ r ) + 1 r 2 bracketleftbig 1 sin ( θ ) ∂ ∂θ ( sin ( θ ) ∂ ∂θ ) + 1 sin 2 ( θ ) ∂ 2 ∂φ 2 bracketrightbig The atom Spherical coordinates Radial wavefunctions n and ℓ Spherical harmonics Selection rules Atom wavefunctions Angular momentum Filling states Spherical coordinates Spherical coordi- nates: φ = azimuthal angle θ = zenith angle r = radius from origin x y z r ( r , θ , φ ) φ θ The atom Spherical coordinates Radial wavefunctions n and ℓ Spherical harmonics Selection rules Atom wavefunctions Angular momentum Filling states The atom II Assume separable variables, as we did with the 2D infinite well (Serway Eq. 8.11): ψ ( r , θ, φ ) = R ( r )Θ( θ )Φ( φ ) The Schrödinger equation then becomes − planckover2pi1 2 2 m braceleftBig ΘΦ r 2 ∂ ∂ r ( r 2 ∂ R ∂ r ) + R r 2 bracketleftbig Φ sin ( θ ) ∂ ∂θ ( sin ( θ ) ∂ Θ ∂θ ) + Θ sin 2 ( θ ) ∂ 2 Φ ∂φ 2 bracketrightbig bracerightBig + U ( r ) R ΘΦ = ER ΘΦ Multiply through by − 2 m planckover2pi1 2 r 2 sin 2 ( θ ) R ΘΦ and rearrange. The atom Spherical coordinates Radial wavefunctions n and ℓ Spherical harmonics Selection rules Atom wavefunctions Angular momentum Filling states The atom III Multiplying through by − 2 m planckover2pi1 2 r 2 sin 2 ( θ ) R ΘΦ and rearranging gives (see Serway Eq. 8.12) 1 Φ d 2 Φ d φ 2 = 2 m planckover2pi1 2 r 2 sin 2 ( θ )( U ( r ) − E ) − sin 2 ( θ ) R d dr ( r 2 dR dr ) − sin ( θ ) Θ d d θ ( sin ( θ ) d Θ d θ ) . Look at this result: the left hand side depends on Φ( φ ) , while the right hand side does not. This must be true for any ( r , θ ) so the left hand side equals a constant! The atom Spherical coordinates Radial wavefunctions n and ℓ Spherical harmonics Selection rules Atom wavefunctions Angular momentum Filling states The atom IV Again, we have 1 Φ d 2 Φ d φ 2 = constant or d 2 Φ d φ 2 − ( constant )Φ = (1) Let’s assume this is associated with the angular momentum projected on the ˆ z axis of L z involving the quantum number m ℓ . In this case we can assume solutions to (see Serway Eq. 8.13) d 2 Φ...
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l15 - The atom Spherical coordinates Radial wavefunctions n...

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