Work Out 2
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This document contains 3 physics problems from textbooks: 1) Prove Green's reciprocity theorem relating potentials due to different charge distributions; 2) Define the partition function and show that taking the derivative as beta approaches infinity gives the ground state energy, illustrated for a particle in a 1D box; 3) Show that geodesics on a spherical surface are great circles through the center of the sphere.
Work Out 2
- 1. Physics Work-out #2 April 30, 2011 Jackson’s Classical Electrodynamics 1.12: Prove Green’s reciprocation theorem: If Φ is the potential due to a volume- charge density ρ within a volume V and a surface-charge desntiy σ on the con- ducting surface S (∂V ) bounding the surface V , while Φ is the potential due to another charge distribution ρ and σ , then ρΦ 3 + σΦ = ρΦ 3 + σΦ V ∂V V ∂V Sakurai’s Modern Quantum Mechanics 2.29: Define the partition function as Z= 3 K( ; 0)|β= / Show that the ground-state energy is obtained by taking 1 ∂Z − (β → ∞) Z ∂β Illustrate this for a particle in a one-dimensional box. Goldstein’s Classical Mechanics 2.4: Show that the geodesics of a spherical surface are great circles, i.e., circles whose centers lie at the center of the sphere. 1