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Unformatted text preview: 1 Physics 316 Solution for homework 6 Spring 2005 Cornell University I. EXERCISE 1 Compute the position uncertainty Δ x in the ground state Φ ( x ) = 1 p √ πa e ξ 2 2 , ξ = x a , a = r ¯ h mω . (1.1) for the harmonic oscillator potential. Use Δ x = p h ( xh x i ) 2 i = q h x 2 ih x i 2 and the expectation values h x i = Z ∞∞ x  Φ ( x )  2 dx , › x 2 fi = Z ∞∞ x 2  Φ ( x )  2 dx . (1.2) (prepared by Steve Drasco) First we will compute the average position h x i h x i = Z ∞∞ dx x  Φ  2 = 0 , (1.3) since x  Φ  2 is an odd function, and the integration boundaries are symmetric about the origin. Here we have used a general rule which applies to any odd function f ( x ) = f ( x ) Z A A dx f ( x ) = Z A dx f ( x ) + Z A dx f ( x ) , = Z A dx f ( x ) + Z A dx f ( x ) , = Z A dx f ( x ) + Z A dx f ( x ) , = 0 . (1.4) We now compute › x 2 fi › x 2 fi = Z ∞∞ dx x 2  Φ  2 , = 1 a √ π Z ∞∞ dx x 2 e Ax 2 , (1.5) where we have defined A = a 2 . Now note that x 2 e Ax 2 = ∂ ∂A e Ax 2 , (1.6) so that we have › x 2 fi = 1 a √ π ∂ ∂A Z ∞∞ dx e Ax 2 , = 1 a ∂ ∂A A 1 / 2 , = a 2 2 , (1.7) 2 where we have used the known Gaussian integral Z ∞∞ dx e Ax 2 = r π A . (1.8) To prove this relation let I = R ∞∞ e Ax 2 . Then we have I 2 = Z ∞∞ dx Z ∞∞ dy e A ( x 2 + y 2 ) , = Z ∞ dr Z 2 π rdφ e Ar 2 , = 2 π Z ∞ dr re Ar 2 , = π/A, (1.9) where in the second line we have converted from cartesian coordinates with area element dxdy , to polar coordinates with area element rdφdr . These results, Eqs. (1.3) and (1.7), give Δ x = q h x 2 ih x i 2 = a √ 2 ≈ . 7 a. (1.10) So on average, measurements should find the particle at the origin, and these measurements should fluctuate on a scale of about 70% of the ground state’s classical maximum extension a . II. EXERCISE 2 What is the expectation value h V i and the uncertainty Δ V = p h V 2 ih V i 2 of the harmonic oscillator potential V = 1 2 mω 2 x 2 for the ground state wave function? This corresponds to the fact that the energy of a classical harmonicfor the ground state wave function?...
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This note was uploaded on 09/28/2008 for the course PHYS 316 taught by Professor Hoffstaetter during the Spring '05 term at Cornell University (Engineering School).
 Spring '05
 HOFFSTAETTER
 Physics, Work

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