851HW11_09Solutions - PHYS851 Quantum Mechanics I, Fall...

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Unformatted text preview: PHYS851 Quantum Mechanics I, Fall 2009 HOMEWORK ASSIGNMENT 11 Topics Covered: Orbital angular momentum, center-of-mass coordinates Some Key Concepts: angular degrees of freedom, spherical harmonics 1. [20 pts] In order to derive the properties of the spherical harmonics, we need to determine the action of the angular momentum operator in spherical coordinates. Just as we have ( x | P x | ) = i planckover2pi1 d dx ( x | ) , we should find a similar expression for ( r | vector L | ) . From vector L = vector R vector P and our knowledge of momentum operators, it follows that ( r | vector L | ) = planckover2pi1 parenleftbigg vectore x parenleftbigg y d dz z d dy parenrightbigg + vectore y parenleftbigg z d dx x d dz parenrightbigg + vectore z parenleftbigg x d dy y d dx parenrightbiggparenrightbigg ( r | ) . Cartesian coordinates are related to spherical coordinates via the transformations x = r sin cos y = r sin sin z = r cos and the inverse transformations r = radicalbig x 2 + y 2 + z 2 = arctan( radicalbig x 2 + y 2 z ) = arctan( y x ) . Their derivatives can be related via expansions such as x = r x r + x + x . Using these relations, and similar expressions for y and z , find expressions for ( r | L x | ) , ( r | L y | ) , and ( r | L z | ) , involving only spherical coordinates and their derivatives. x r = x r = sin cos x = z 2 r 2 x z x 2 + y 2 = cos cos r x = x 2 x 2 + y 2 y x 2 = csc sin r So d dx = sin cos r + cos cos r csc sin r y r = y r = sin sin y = z 2 r 2 y z x 2 + y 2 = cos sin r y = x 2 x 2 + y 2 1 x = csc cos r So d dy = sin sin r + cos sin r + csc cos r z r = z r = cos 1 z = z 2 r 2 x 2 + y 2 z 2 = sin r z = 0 So d dz = cos r sin r Now ( r | L x | ) = i planckover2pi1 ( y d dz z d dy ) ( r | ) So we can say L x = i planckover2pi1 parenleftbigg y d dz z d dy parenrightbigg = i planckover2pi1 ( r sin cos sin r sin 2 sin r sin cos sin r cos 2 sin + cot cos ) Which means ( r | L x | ) = i planckover2pi1 ( sin cot cos ) ( r | ) Similarly we can say L y = i planckover2pi1 ( z d dx x d dz ) = i planckover2pi1 ( r sin cos cos r + cos 2 cos cot sin r sin cos cos r + sin...
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851HW11_09Solutions - PHYS851 Quantum Mechanics I, Fall...

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