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The hydrogen atom radial Hamiltonian 1 ə Ә 2r² ar -2 Ĥ = = can be written in atomic units as l(l + 1) 1 1 a 2r² r 2r² ər 1 √2π + with its lowest eigenvalue, E100 100 (r, 0.4) er. In the last equality for F, we have substituted f = 0 (s-orbital). (a) Use a Gaussian function, (r) = e-ar²/2, to represent this lowest eigenstate and find an expression for the lowest energy level as a function of the parameter, a. Note that Sºx xe-Br2 50⁰: xe-Bx2, ²017) - 1²/3 ar 1 == -, corresponding to the normalized eigenfunction, dx = 2/13 √ x²e-³x² dx = 2ß 4 -2 TT B3 dx = 3 TL 8 B5

The hydrogen atom radial Hamiltonian 1 ə Ә 2r² ar -2 Ĥ = = can be written in atomic units as l(l + 1) 1 1 a 2r² r 2r² ər 1 √2π + with its lowest eigenvalue, E100 100 (r, 0.4) er. In the last equality for F, we have substituted f = 0 (s-orbital). (a) Use a Gaussian function, (r) = e-ar²/2, to represent this lowest eigenstate and find an expression for the lowest energy level as a function of the parameter, a. Note that Sºx xe-Br2 50⁰: xe-Bx2, ²017) - 1²/3 ar 1 == -, corresponding to the normalized eigenfunction, dx = 2/13 √ x²e-³x² dx = 2ß 4 -2 TT B3 dx = 3 TL 8 B5

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The hydrogen atom radial Hamiltonian can be written in atomic units as 1_ª (r² - ) 2r² Ĥ = 1 √2π dx = 1 2ß 1 a 2r² ər 1 -, corresponding to the normalized eigenfunction, with its lowest eigenvalue, E100 100 (r, 0.4) = er. In the last equality for F, we have substituted f = 0 (s-orbital). (a) Use a Gaussian function, (r) = e-ar²/2, to represent this lowest eigenstate and find an expression for the lowest energy level as a function of the parameter, a. Note that Sººx xe-Br2 50⁰0 3 x¹e-³x² dx 3 T 8 l(l + 1) 1 2r² r + == √° x²e-³x² dx = 4 2 ²017) - 1²/3 ar TL B3 = B5
Transcribed Image Text:The hydrogen atom radial Hamiltonian can be written in atomic units as 1_ª (r² - ) 2r² Ĥ = 1 √2π dx = 1 2ß 1 a 2r² ər 1 -, corresponding to the normalized eigenfunction, with its lowest eigenvalue, E100 100 (r, 0.4) = er. In the last equality for F, we have substituted f = 0 (s-orbital). (a) Use a Gaussian function, (r) = e-ar²/2, to represent this lowest eigenstate and find an expression for the lowest energy level as a function of the parameter, a. Note that Sººx xe-Br2 50⁰0 3 x¹e-³x² dx 3 T 8 l(l + 1) 1 2r² r + == √° x²e-³x² dx = 4 2 ²017) - 1²/3 ar TL B3 = B5
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(b) What is the variational energy for the optimized trial function? Compare this energy to the exact value (-2), and evaluate the percent error of your value: Evar - Eexact Eexact x 100%
Transcribed Image Text:(b) What is the variational energy for the optimized trial function? Compare this energy to the exact value (-2), and evaluate the percent error of your value: Evar - Eexact Eexact x 100%
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