851HW10_09 - PHYS851 Quantum Mechanics I Fall 2009 HOMEWORK...

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PHYS851 Quantum Mechanics I, Fall 2009 HOMEWORK ASSIGNMENT 10 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 | rθφ ) , where vector R | rθφ ) = rvectore r ( θ,φ ) | rθφ ) . In cartesian coordinates, the same state is written | xyz ) , where vector R | xyz ) = ( xvectore x + yvector e y + zvectore z ) | xyz ) . Evaluate the following: (a) ( rθφ | r θ φ ) (b) ( rθφ | xyz ) (c) ( rθφ | p x p y p z ) (d) ( rθφ | vector R | r θ φ ) (e) ( rθφ | Z | r θ φ ) (f) ( rθφ | P z | r θ φ ) 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 . (a) Write the quantum mechanical Hamiltonian that describes this system. (b) Define a suitable tensor-product-state basis to describe the system.

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