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Problem Set 7

# Problem Set 7 - X ﬂuc t = t = 0 Show f t = X l e i S l...

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PHYSICS 230 – PROBLEM SET 7 These problems will become clearer after the lecture on Monday. 1. Shankar Problem 8.6.2 2. Consider a free particle of mass m on a circle of radius L described by a coordinate x running from 0 to 2 πL . We will evaluate f ( t 0 ) ≡ h x = 0 | U ( t 0 ) | x = 0 i in two ways ( t 0 > 0). (a)Operator Method: Inserting a complete set of the energy eigenbasis, f ( t 0 ) = h x = 0 | e - i ˆ Ht 0 ¯ h ( X n | n ih n | ) | x = 0 i = X n a n ( t 0 ) . (1) (Remember the solution of a free particle with periodic boundary con- ditions in set 2.) Calculate a n ( t 0 ). (b)Path Integral Method: Find all the classical paths here. Now show we can express a path connecting the same point on the circle as X ( t ) = 2 πlLt t 0 + X fluc ( t ), where l Z is a “winding number” and X fluc : [0 , t 0 ] R is a continuous function with X fluc ( t = 0) =

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Unformatted text preview: X ﬂuc ( t = t ) = 0. Show f ( t ) = X l e i S ( l ) winding ( t ) ¯ h × Z [ D X ﬂuc ] e i S ¯ h = c ( t ) X l e i S ( l ) winding ( t ) ¯ h . (2) Calculate S ( l ) winding ( t ) and c ( t ). In obtaining the latter observe that this is just the same result needed to correct the classical result for the free particle on the real line R . (c)Show that the results in part (a) and (b) are equivalent by using the Poisson resummation formula ∞ X n =-∞ e-απ 2 n 2 = 1 √ πα ∞ X l =-∞ e-l 2 α . (3) (d) Prove f ( t ) is periodic in time. This is diﬀerent than the real line case....
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Problem Set 7 - X ﬂuc t = t = 0 Show f t = X l e i S l...

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