Use a trial function ψ=re-αr , with α as a variational parameter to calculate the ground state energy of the H atom. I t

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Use a trial function ψ=re-αr , with
α as a variational parameter to calculate the ground state
energy of the H atom.
I tried doing this question by following the hints given in the
question. I attached my workings as I'm not sure about them so can
someone check if they're right? If they're wrong, I would greatly
appreciate a solution that's found by following the hints given in
the question.
I saw some answers online that showed 0.0046 hartee
and -13.6eV. I'm not sure which is right

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= Problem 4 Use a trial function y = re-ar, with a as a variational parameter to calculate the ground state energy of the H atom. Hints: 1) Normalize the wave function using spherical coordinate system 2) Find the expression for the energy as function of parameter a. Use the Hamiltonian operator as for H atom, easiest is to employ atomic units: 1/a2 2 a À = Ñ + Û = 1(1+1) 1 2 Or2 p2 Assuming that the ground state is a 1s orbital – the l = 0. This leads to a smaller Hamiltonian: H = R + V = 2 a 2 rar 3) Evaluate the derivative of energy with respect to parameter a. Determine the best value of a by solving the equation: = 0. 4) Write down the wave function and energy for the obtained value of a. +-- + rar r = 1 (0* + ) - r dB da
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