# solution1 - 1 Consider a particle of mass m moving in 1D...

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1. Consider a particle of mass m moving in 1D subject to the con f ning potential V ( x )= k | x | , where k is a positive constant. (a) Using a trial wave function of the form φ α ( x α exp ( α | x | ) , we compute 1 2 m h P 2 i α = ~ 2 2 m Z dx | φ 0 | 2 = ~ 2 α 2 2 m and h V i α = k Z dx | x || φ | 2 =2 k α Z 0 dx xe 2 α x = k 2 α Z 0 ue u du = k 2 α so that h H i α = ~ 2 α 2 2 m + k 2 α . Setting h H i α ∂α =0= 2 ~ 2 α 3 km m α 2 we deduce that ~ 2 α 2 2 m = k 4 α and thus that α = μ 2 ~ 2 1 / 3 . From this value of α , we obtain the following estimate (upper bound) for the ground state energy: E var 0 = ~ 2 α 2 2 m + k 2 α = 3 k 4 α = 3 2 μ ~ 2 k 2 4 m 1 / 3 0 . 944 μ ~ 2 k 2 m 1 / 3 . (b) Anticipating that the f rst excited state will be an odd function of x (and thus orthogonal to the ground state) we consider the normalized wave function φ ( x φ 1 α ( x 2 α 3 / 2 x exp ( α | x | ) for which φ ( 0 x ∂φ x = 2 α 3 2 e α | x | (1 α | x | ) so that 1 2 m h P 2 i α = ~ 2 2 m Z −∞ dx ¯ ¯ φ 0 ¯ ¯ 2 = ~ 2 2 m 2 α 3 Z

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solution1 - 1 Consider a particle of mass m moving in 1D...

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