# Alternative Formulations for Angular Momentum Operators, Cartesian and Spherical Polar Forms

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Alternative Formulations for Angular Momentum Operators, Cartesian and Spherical Polar Forms University of Connecticut [email protected] Chemistry Education Materials Department of Chemistry 8-1-2006 Alternative Formulations for Angular Momentum Operators, Cartesian and Spherical Polar Forms Carl W. David University of Connecticut, [email protected] Follow this and additional works at: http://digitalcommons.uconn.edu/chem_educ This Article is brought to you for free and open access by the Department of Chemistry at [email protected] It has been accepted for inclusion in Chemistry Education Materials by an authorized administrator of [email protected] For more information, please contact [email protected] Recommended Citation David, Carl W., "Alternative Formulations for Angular Momentum Operators, Cartesian and Spherical Polar Forms" (2006). Chemistry Education Materials. Paper 18. http://digitalcommons.uconn.edu/chem_educ/18 Alternative Formulations for Angular Momentum Operators, Cartesian and Spherical Polar Forms C. W. David Department of Chemistry University of Connecticut Storrs, Connecticut 06269-3060 (Dated: August 1, 2006) I. CARTESIAN AND SPHERICAL POLAR FORMS It is of value to inspect the angular momentum opera- tor in terms of angles rather than Cartesian coo¨rdinates. Remember that x = r sinϑ cosϕ y = r sinϑ sinϕ z = r cosϑ and ~L = iˆ jˆ kˆx y z px py pz = iˆ(ypz − zpy) + jˆ(zpx − xpz) + kˆ(xpy − ypx) (1.1) We start with a feast of partial derivatives:( ∂r ∂x ) y,z = sinϑ cosϕ (1.2) ( ∂r ∂y ) x,z = sinϑ sinϕ (1.3) ( ∂r ∂z ) x,y = cosϑ (1.4) ( ∂ϑ ∂x ) y,z = cosϑ cosϕ r (1.5) ( ∂ϑ ∂y ) x,z = cosϑ sinϕ r (1.6) ( ∂ϑ ∂z ) x,y = − sinϑ r (1.7) and finally ( ∂ϕ ∂x ) y,z = − sinϕ r sinϑ (1.8) ( ∂ϕ ∂y ) x,z = cosϕ r sinϑ (1.9) ( ∂ϕ ∂z ) x,y = 0 (1.10) which we employ on the defined x-component of the an- gular momentum, thus L

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