851HW10_09Solutions - PHYS851 Quantum Mechanics I Fall 2009...

Info iconThis preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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 ....
View Full Document

Page1 / 7

851HW10_09Solutions - PHYS851 Quantum Mechanics I Fall 2009...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online