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# pset2sol - 22.02 Problem Set 2 Solution 1 Particle in 1D...

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22.02 Problem Set 2 Solution 1. Particle in 1D box a. The Eigenvalue equation within the box is: n n n n E x m H ϕ ϕ ϕ = 2 2 2 2 ˆ h 0 ) ( = x n 0 ) ( ) 0 ( = = L n n ϕ ϕ = ( 1 . 1 ) Since ϕ outside the box and it is a continuous function, it should satisfy the flowing boundary conditions: ( 1 . 2 ) Rewrite (1.1) as: 0 2 2 2 = + n n n k x ϕ ϕ ( 1 . 3 ) 2 2 2 h n n mE k = x k B x k A n n n cos sin + = ϕ A x B n n ) ( 0 ) 0 ( = = ϕ ϕ L k L k A L n n n = = 0 sin ) ( A ) ( x n ( 1 . 4 ) The solution to (1.3) is: ( 1 . 5 ) Use boundary conditions in (1.2), we get: , and (1.6) , (1.7) To get , we normalize ϕ as: x k n sin π n = ,... 2 , 1 , 0 = n dx L n 0 0 2 ϕ L L x n 0 2 sin π = A xdx 2 = A dx x 2 ) ( L n k 2 sin 1 2 2 2 cos 1 0 2 = = π θ θ n L A d sin 2 2 = π π θ θ n L A d sin 0 2 π L L n 0 2 = π n n L A 2 π d x n L A π L x n L 2 = A ( 1 . 8 ) Put (1.5), (1.6), (1.7) and (1.8) together, we get the Eigenfunctions as: = L x n L n π ϕ sin 2 0 ) ( = x n ϕ 0 < x , , n , or The energy spectrum ’s are: L = x 0 L x > n E ,... 2 , 1 , 0 2 2 2 mL h = n 2 2 2 2 2 n m k E n n h π = = , ,... 2 , 1 , 0 b.

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pset2sol - 22.02 Problem Set 2 Solution 1 Particle in 1D...

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