Question

45. Use a Gaussian trial function o(r) = e" to compute the ground state energy of the hydrogen atom. What is the "best" value of the variational parameter a? How far off is this energy from the true value?

Answer 1

using a Gaussian trial as trial function for hydrogen atom ground state. This problem could be solved by the variational method by obtaining the energy of $60) as a function of variational parameter as and the minimizing E(&) to find the optimum value amin. Onblet te 7 The variational theavcem's approximate wave function & energy of per) as functionie $() = &- amino and Elamin) in Frequently, the trial function is written as a linear combination of bails functions, such as $ - Ecid: 1027 10 let This leads to linear variation method and parameters are the expansion coefficients- variational - li. Theme energy for approx wavefundfon Ich 4 cores se FO) 1 Eccs so =

His simplification SPA 0; 2 mentes & can be use for Jodoh di J. Si s ob s ccHis id we get Eloi - CC His tci Sis - Diffrentiation of ② worto the expansion coefficient. Ci Vields a non-Trivial solution Only if "secular determinant "equals o IH, - ES, H, ESa Hip - Es ... Hin - ESIN Her - ES, 2... Han - ESon Hall - ESNI HH2-ESHO -.. Han-ESNN If an orthonormal equation is greatly for i=j & o for basis in used , the secular simplified because Sie ist itj . HI-E H 2 H22 - E - - .. H₂i HAN HI Holz

In either case, the secular determinar for N. basis function gives and the order polynomial in E which is solved for N different rook, each of which approx a different eigen- value. SON