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lecture_11_07b

lecture_11_07b - Particle states in the square well n=3...

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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 0 L x
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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.
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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 -->
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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 ) 0 L = 2 L L np 3 dJ J sin J ( ) 2 0 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...
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05/12/09 Physics 13 - Fall 07 - G.R. Goldstein
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