PrtinboxHeisenbergn

# PrtinboxHeisenbergn - Particle-in-a-Box and Heisenberg...

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Colby College Particle-in-a-Box and Heisenberg Uncertainty <x> = a / 2 <p> = 0 σ x σ p h - / 2 σ x = (<x 2 > – <x> 2 ) ½ σ p = (<p 2 > – <p> 2 ) ½ p ^ 2 = p ^ p ^ = – h - 2 (d 2 /dx 2 ) Ψ 1 = 2 a ½ sin(n π x/a) 0 2 4 6 8 10 12 0 2 4 6 8 10 E n /(h 2 /8ma ) x (Å) <x 2 > = a o Ψ * n x 2 Ψ n dx a o Ψ * n Ψ n dx = a o x 2 Ψ 2 n dx = 2 a a o x 2 sin 2 (n π x/a) dx Change in variables: y = n π x/a, dy/dx = n π /a, dx = a/(n π ) dy, x = a/(n π ) y : <x 2 > = 2 a a n π 3 0 π y 2 sin 2 (y) dy 0 π x 2 sin 2 (x) dx = x 3 6 x 2 4 1 8 sin 2x – x cos 2x 4 π 0 = (n π ) 3 6 n π 4 sin 2x = sin(2n π ) = 0 , sin 0 = 0 , cos 2x = cos(2n π ) = 1 : <x 2 > = 2 a a n π 3 (n π ) 3 6 n π 4 = a 2 1 3 1 2n 2 π 2 σ 2 x = <x 2 > – <x> 2 = a 2 1 3 1 2n 2 π 2 a 2 4 = a 2 12 a 2 2n 2 π 2 = a 2 4n 2 π 2 n 2 π 2 3 – 2 σ x = a 2n π n 2 π 2 3 – 2 ½ <p 2 > = a o Ψ * n p ^ 2 Ψ n dx a o Ψ * n Ψ n dx = h - 2 0 a Ψ n d 2 dx 2 Ψ n dx = – h - 2 2 a
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