notes06-07-page230

notes06-07-page230 - The Laplacian 2 = 2 2 2 + 2+ 2 x2 y z...

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The Laplacian 2 = μ 2 ∂x 2 + 2 ∂y 2 + 2 ∂z 2 . (31.19) Normalized wavefunctions for the 3D particle in a box, ψ n ( x )= 2 2 abc sin n x πx a sin n y πy b sin n z πz c . (31.20) Theenergyleve lsforthe3Dpart ic leinabox , E n x ,n y ,n z = n 2 x h 2 8 ma 2 + n 2 y h 2 8 mb 2 + n 2 z h 2 8 mc 2 . (31.21) Orthonormality: Z space ψ j ψ k = ( 1 ,j = k 0 ,j 6 = k . (31.22) Superpostion: ψ = X i a i ϕ i (31.23) Commonly used comutators of the angular momentum operators are h ˆ L x , ˆ L y i = i ~ ˆ L z , h ˆ L y , ˆ L z i = i ~ ˆ L x , h ˆ L z , ˆ L
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