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Unformatted text preview: 05/12/09 Physics 13  Fall 07  G.R. Goldstein 1 Particle states in the square well E 1 E 2 =4E 1 E 3 =9E 1 n=1 n=2 n=3 For probability in finite interval ( x 1 , x 2 ) dxy n ( x ) 2 x 1 x 2 L x 05/12/09 Physics 13  Fall 07  G.R. Goldstein 2 Where is the particle? ψ n ( x ) 2 dx = 2 L sin 2 ( npx L ) dx or 1 cos 2 npx L L dx Greatest probability to be where density is maximum: center or x=L/2 for n=1; at x=L/4 and 3L/4 for n=2; at x=L/6, L/2,5L/6 for n=3; etc. But also places with 0 probability for each n. Probability “clouds” within box. 05/12/09 Physics 13  Fall 07  G.R. Goldstein 3 Implications of square well solutions • Particle is not anywhere in particular  it is distributed  cloud • Large n approaches uniform distribution e.g. n=12 Bohr’s Correspondence principle QM > Classical as n > ∞ 05/12/09 Physics 13  Fall 07  G.R. Goldstein 4 Average position in square well Always have x n = L 2 but x 2 n will depend on n and r.m.s. deviation x 2 n x n 2 varies with n. x 2 n = dx x 2 2 L sin 2 ( npx / L ) L = 2 L L np 3 dJ J sin J ( ) 2 np = L 2 1 3 1 2 n 2 p 2 so D x rms = L 1 3 1 2 n 2 p 2 1 4 = L 1 12 1 2 n 2 p 2 Expectation value of x 2 or average over... 05/12/09 Physics 13  Fall 07  G.R. GoldsteinPhysics 13  Fall 07  G....
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This note was uploaded on 03/27/2008 for the course PHY 13 taught by Professor Garyr.goldstein during the Fall '04 term at Tufts.
 Fall '04
 GaryR.Goldstein
 Physics

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