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Unformatted text preview: PHYS851 Quantum Mechanics I, Fall 2009 HOMEWORK ASSIGNMENT 10: Solutions Topics Covered: Tensor product spaces, change of coordinate system, general theory of angular mo- mentum Some Key Concepts: Angular momentum: commutation relations, raising and lowering operators, eigenstates and eigenvalues. 1. [10 pts] Consider the position eigenstate | vector r ) . In spherical coordinates, this state is written as | r ) , where vector R | r ) = rvectore r ( , ) | r ) . In cartesian coordinates, the same state is written | xyz ) , where vector R | xyz ) = ( xvectore x + yvector e y + zvectore z ) | xyz ) . Evaluate the following: Most of these can be evaluated in many different ways, I am just giving one possibility for each: (a) ( r | r ) = 1 r 2 sin ( r r ) ( ) ( ) (b) ( r | xyz ) = ( r sin cos x ) ( r sin sin y ) ( r cos z ) (c) ( r | p x p y p z ) = [2 planckover2pi1 ] 3 / 2 exp bracketleftbig i planckover2pi1 ( p x r sin cos + p y r sin sin + p z r cos ) bracketrightbig (d) ( r | vector R | r ) = 1 r sin vector e r ( , ) ( r r ) ( ) ( ) (e) ( r | Z | r ) = cot r vector e r ( , ) ( r r ) ( ) ( ) (f) ( r | P z | r ) = i planckover2pi1 z ( r | r ) now z = r z r + z + z = sec r 1 r sin so that ( r | P z | r ) = 2 i planckover2pi1 sec csc r 3 ( r r ) ( ) ( ) i planckover2pi1 sec csc r 2 ( r r ) ( ) ( ) i planckover2pi1 cos csc 3 r 3 ( r r ) ( ) ( ) + i planckover2pi1 csc 2 r 3 ( r r ) ( ) ( ) 1 2. [10 pts] Consider a system consisting of two spin-less particles with masses m 1 and m 2 , and charges q 1 and q 2 ....
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