5210-chap6

# 5210-chap6 - Chapter 6 Slide 1 Chapter 6 The Hydrogen Atom...

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Unformatted text preview: Chapter 6: Slide 1 Chapter 6 The Hydrogen Atom Chapter 4: Slide 2 The Three Dimensional Schrödinger Equation In Cartesian Coordinates, the 3D Schrödinger Equation is: ψ ψ ψ ψ ψ E z y x V z m y m x m = + ∂ ∂- ∂ ∂- ∂ ∂- ) , , ( 2 2 2 2 2 2 2 2 2 2 2 2 h h h ) , , ( z y x ψ ψ = The Laplacian in Cartesian Coordinates is: 2 2 2 2 z y x ∂ ∂ + ∂ ∂ + ∂ ∂ ≡ ∇ T(x) T(y) T(z) V(x,y,z) Therefore: ψ ψ E z y x V m = + ∇- ) , , ( 2 2 2 h It is sometimes not possible to solve the Schrödinger exactly in Cartesian Coordinates (e.g. the Hydrogen Atom), whereas it can be solved in another coordinate system. The “Rigid Rotor” and the Hydrogen Atom can be solved exactly in Spherical Polar Coordinates. Chapter 4: Slide 3 Spherical Polar Coordinates To specify a point in space requires three coordinates. In the spherical polar coordinate system, they are: r 0 ≤ r < ∞ Distance of point from origin (OP) θ 0 ≤ θ < π Angle of vector (OP) from z-axis φ 0 ≤ φ < 2 π Angle of x-y projection (OQ) from x-axis Chapter 4: Slide 4 Relation of Cartesian to Spherical Polar Coordinates θ r z r z = θ ) cos( ) cos( θ = r z x-axis y-axis O Q ϕ x y OQ x = ) cos( φ ) cos( φ OQ x = ) cos( ) sin( φ θ r x = OQ y = ) sin( φ ) sin( φ OQ y = ) sin( ) sin( φ θ r y = OQ=rsin( θ ) OP Chapter 4: Slide 5 The Volume Element in Spherical Polar Coordinates In Cartesian Coordinates, the volume element is: dxdydz dV = In spherical polar coordinates, the volume element is: φ θ OQd rd dr dV • • = φ θ θ d r rd dr dV ) sin( • • = φ θ θ d d dr r dV ) sin( 2 = Chapter 4: Slide 6 The Laplacian in Spherical Polar Coordinates 2 2 2 2 z y x ∂ ∂ + ∂ ∂ + ∂ ∂ ≡ ∇ Cartesian Coordinates: One example of a chain rule formula connecting a derivative with respect to x, y, z to derivatives with respect to r, θ , φ is: x x x r r x ∂ ∂ ∂ ∂ + ∂ ∂ ∂ ∂ + ∂ ∂ ∂ ∂ = ∂ ∂ φ φ θ θ It may be shown that by repeated application of chain rule formulae of this type (with 2-3 hours of tedious algebra), the Laplacian in spherical polar coordinates is given by: 2 2 2 2 2 2 2 2 ) ( sin 1 ) sin( ) sin( 1 1 φ θ θ θ θ θ ∂ ∂ + ∂ ∂ ∂ ∂ + ∂ ∂ ∂ ∂ ≡ ∇ r r r r r r Chapter 4: Slide 7 Angular Momentum Operators in Spherical Polar Coordinates ∂ ∂- ∂ ∂- = x y y x i L z z ˆ It may be shown that φ ∂ ∂ = φ ∂ ∂- = i i L z z z ˆ z z y y x x L L L L L L L ˆ ˆ ˆ ˆ ˆ ˆ • + • + • = 2 ^ It may be shown that φ ∂ ∂ θ + θ ∂ ∂ θ θ ∂ ∂ θ- = 2 2 2 2 2 1 1 ) ( sin ) sin( ) sin( c L ^ Chapter 4: Slide 8 Outline • Math Preliminary: Products of Vectors • Rotational Motion in Classical Physics • The 3D Quantum Mechanical Rigid Rotor • Angular Momentum in Quantum Mechanics • Angular Momentum and the Rigid Rotor • The 2D Quantum Mechanical Rigid Rotor...
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5210-chap6 - Chapter 6 Slide 1 Chapter 6 The Hydrogen Atom...

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