Unformatted text preview: as a ψ  H  ψ A , where according to chapter 3.5.3, the Hamiltonian is H = − ¯ h 2 2 m ∂ 2 ∂x 2 Compare the ground state energy with the exact value, E 1 = ¯ h 2 π 2 / 2 mℓ 2 (Hints: i ℓ x ( ℓ − x ) d x = ℓ 3 / 6 and i ℓ x 2 ( ℓ − x ) 2 d x = ℓ 5 / 30) Answer: To satisfy the normalization requirement that the particle must be somewhere, you need a ψ  ψ A = 1, or substituting for ψ , 1 = a ax ( ℓ − x )  ax ( ℓ − x ) A =  a  2 a x ( ℓ − x )  x ( ℓ − x ) A And by de±nition, chapter 2.3, the ±nal inner product is just the integral i ℓ x 2 ( ℓ − x ) 2 d x which is given to be ℓ 5 / 30. So you must have  a  2 = 30 ℓ 5...
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 Fall '11
 Dr.DanielArenas
 mechanics, 2m, Ψ, 4.5.6.1, hydrogen molecular ion, normalization requirement

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