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PS1_solution - MECE 6700 Carbon Nanotube Sci Tech James...

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James Hone, Fall 2011 Problem Set 1 Due on Sept. 21 th 1 . For an 1D particle in a box, calculate: h x 2 i = Z Ψ * x 2 Ψ dx h p 2 i = - ~ 2 Z Ψ * 2 ∂x 2 Ψ dx for the two lowest-energy eigenstates. Then calculate Δ x Δ p = p h x 2 i - h x i 2 · p h p 2 i - h p i 2 and compare to the minimum set by the Heisenberg principle. Note : It was given that Δ x Δ p = p h x 2 i · p h p 2 i , but it is only valid when h x i = h p i = 0, that requires the 1D particle is bounded in a box symmetrical about original, i.e. from - a/ 2 to a/ 2. A general deﬁnition is Δ x Δ p = p h ( x - h x i ) 2 i · p h ( p - h p i ) 2 i , and will lead to the equation above. Solution The lowest energy eigenstate is give as Ψ 1 = r 2 a sin( π a x ) , for the 1D box deﬁned from 0 to a . Then h x 2 i = Z a 0 ( 2 a ) x 2 sin 2 ( π a x ) dx = a 2 (2 π 2 - 3) 6 π 2 , h p 2 i = - ~ 2 Z a 0 ( 2 a )sin( π a x ) 2 ∂x 2 sin( π a x ) dx = 2 π 2 ~ 2 a 3 Z a 0 sin 2 ( π a x

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PS1_solution - MECE 6700 Carbon Nanotube Sci Tech James...

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