Chapter4_4 - PHY4604 R D Field Spherical Coordinates...

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Unformatted text preview: PHY4604 R. D. Field Spherical Coordinates Spherical Coordinates: We must express the operators in spherical coordinates where z r = x2 + y2 + z2 x = r sin θ cos φ cos θ = z / r = z / x 2 + y 2 + z 2 y = r sin θ sin φ z = r cos θ tan φ = y / x Hence, ∂r ∂r ∂r = sin θ cos φ = sin θ sin φ = cos θ ∂x ∂y ∂z θ rcosθ r rsinθsinφ rsinθcosφ φ rsinθ x 1 ∂θ 1 ∂θ 1 ∂θ = cos θ cos φ = cos θ sin φ = − sin θ . r ∂x r ∂y r ∂z ∂φ 1 sin φ ∂φ 1 cos φ ∂φ =− = =0 r sin θ ∂y r sin θ ∂z ∂x Chain Rule: Use the chain rule. For example, ∂ f ∂f ∂r ∂f ∂θ ∂f ∂ φ = + + . ∂x ∂r ∂x ∂θ ∂ x ∂φ ∂x and thus ∂ ∂1 ∂ 1 sin φ ∂ ( p x ) op = −ih = −ih sin θ cos φ + cos θ cos φ − ∂x ∂r r ∂θ r sin θ ∂φ ∂ ∂1 ∂ 1 cos φ ∂ ( p y ) op = −ih = −ih sin θ sin φ + cos θ sin φ + ∂y ∂r r ∂θ r sin θ ∂φ ( p z ) op = −ih ∂ ∂1 ∂ = −ih cos θ − sin θ ∂x ∂r r ∂θ and ∂ ∂ ∂ ∂ ( Lx )op = −ih y − z = ih sin φ + cot θ cos φ ∂z ∂y ∂θ ∂φ ∂ ∂ ∂ ∂ ( Ly )op = −ih z − x = ih − cos φ + cot θ sin φ ∂θ ∂φ ∂z ∂x ∂ ∂ ∂ ( Lz )op = −ih x − y = −ih ∂y ∂x ∂φ Department of Physics Chapter4_4.doc University of Florida y ...
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This note was uploaded on 05/29/2011 for the course PHY 4064 taught by Professor Fields during the Spring '07 term at University of Florida.

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