851HW10_09Solutions

# 851HW10_09Solutions - PHYS851 Quantum Mechanics I Fall 2009 HOMEWORK ASSIGNMENT 10 Solutions Topics Covered Tensor product spaces change of

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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 | 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: Most of these can be evaluated in many different ways, I am just giving one possibility for each: (a) ( rθφ | r ′ θ ′ φ ′ ) = 1 r 2 sin θ δ ( r − r ′ ) δ ( θ − θ ′ ) δ ( φ − φ ′ ) (b) ( rθφ | xyz ) = δ ( r sin θ cos φ − x ) δ ( r sin θ sin φ − y ) δ ( r cos θ − z ) (c) ( rθφ | p x p y p z ) = [2 π planckover2pi1 ] − 3 / 2 exp bracketleftbig i planckover2pi1 ( p x r sin θ cos φ + p y r sin θ sin φ + p z r cos θ ) bracketrightbig (d) ( rθφ | vector R | r ′ θ ′ φ ′ ) = 1 r sin θ vector e r ( θ,φ ) δ ( r − r ′ ) δ ( θ − θ ′ ) δ ( φ − φ ′ ) (e) ( rθφ | Z | r ′ θ ′ φ ′ ) = cot θ r vector e r ( θ,φ ) δ ( r − r ′ ) δ ( θ − θ ′ ) δ ( φ − φ ′ ) (f) ( rθφ | P z | r ′ θ ′ φ ′ ) = − i planckover2pi1 ∂ z ( rθφ | r ′ θ ′ φ ′ ) now ∂ z = ∂r ∂z ∂ r + ∂θ ∂z ∂ θ + ∂φ ∂z ∂ φ = sec θ∂ r − 1 r sin θ ∂ θ so that ( rθφ | P z | r ′ θ ′ φ ′ ) = 2 i planckover2pi1 sec θ csc θ r 3 δ ( r − r ′ ) δ ( θ − θ ′ ) δ ( φ − φ ′ ) − i planckover2pi1 sec θ csc θ r 2 δ ′ ( r − r ′ ) δ ( θ − θ ′ ) δ ( φ − φ ′ ) − i planckover2pi1 cos θ csc 3 θ r 3 δ ( r − r ′ ) δ ( θ − θ ′ ) δ ( φ − φ ′ ) + i planckover2pi1 csc 2 θ r 3 δ ( r − r ′ ) δ ′ ( θ − θ ′ ) δ ( φ − φ ′ ) 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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## This note was uploaded on 12/04/2011 for the course PHY 7070 taught by Professor Smith during the Spring '11 term at Wisconsin.

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851HW10_09Solutions - PHYS851 Quantum Mechanics I Fall 2009 HOMEWORK ASSIGNMENT 10 Solutions Topics Covered Tensor product spaces change of

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