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Unformatted text preview: PHYSICS 230 – PROBLEM SET 8 1. Consider a particle of mass m conﬁned on a thin ring of circumference L, through which there is a magnetic ﬂux Φ. We can model this system by the Hamiltonian 1 e2 H= p− A (1) 2m c where A = in L).
Φ L = constant and x x + L (i.e., wavefunctions are periodic (a) Sketch how the spectrum of H changes as a function of Φ. Express Tr[exp(− Hτ )] as a summation over energy eigenstates. Conﬁrm that it h ¯ h is invariant under Φ → Φ + 2πe¯ c . (b)Express Tr[exp(− Hτ )] as a summation over periodic ﬁeld conﬁguh ¯ rations, and then doing the gaussian decomposition into a sum over winding numbers. You may use the following path integral expression based on the Lagrangian presented in class: Tr exp − where
τ Hτ h ¯ = 1 [Dx]periodic exp − SE [x(tE )] h ¯ m (2) SE [x(tE )] = 0 dtE 2 dx dtE 2 eA −i c dx . dtE 2π ¯ c h . e (3) Again, conﬁrm that it is invariant under Φ → Φ + (c)Show that the results in part (a) and (b) are equivalent by using the Poisson resummation formula
n=−∞ −απ 2 (n−γ )2 ∞ l2 1 e− α +2πiγl . =√ πα l=−∞ (4) 2. Think about how you would do problem 2.28 in Sakurai (new edition numbering). You do not have to do it! 3. Sakurai problem 2.39 (new edition numbering). You may have to read some of this chapter to do it. ...
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- Spring '09