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Unformatted text preview: Quantum Mechanics Problem Sheet 8 (Solutions are to be handed in for marking before the lecture on Monday, 14 Mar 2005) 1. The stationary Schr¨ odinger equation for a particle moving in a central potential V ( r ) is E Φ( r, θ, φ ) =- ¯ h 2 2 m ∂ 2 Φ ∂r 2 + 2 r ∂ Φ ∂r + 1 2 mr 2 ˆ L 2 Φ( r, θ, φ ) + V ( r )Φ( r, θ, φ ) , where ˆ L is the angular momentum operator for the particle’s motion. (a) Write the wave function Φ( r, θ, φ ) as a product of a radial function R ( r ) and an angular momentum eigenstate Y ‘m ( θ, φ ), and derive the differential equation for R ( r ). (b) In the resulting equation, substitute R ( r ) = F ( r ) /r and derive the differential equation that F ( r ) must satisfy. How does the equation for F ( r ) compare to the stationary Schr¨ odinger equation for a particle moving in one dimension? 2. Consider a particle in a one-dimensional “box” with sagging bottom V ( x ) =- V sin( πx/L ) for 0 ≤ x ≤ L ∞ for x < 0 and x > L .x > L ....
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