Use the variational trial function Ψ Y 10 r exp cr 2 to calculate the total

Use the variational trial function ψ y 10 r exp cr 2

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The non-linear variational method applied to the hydrogen atom. Use the variational trial function: Ψ = Y 10 r exp cr 2 ( ) to calculate the total energy of the hydrogen atom via the following steps: (a) From your knowledge of the exact solutions of the hydrogen atom, explain which state this trial function will be modeling, and why.
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(b) Obtain an expression for the energy in terms of the variational parameter, c . Make a plot of the energy versus c for values of c that range between obviously too small (e.g. starting with 0.1), and obviously too large (e.g. ending with 10), adjusting so that the plot provides useful visual information. (c) Find the optimum value of the parameter c that minimizes the energy, and the associated energy, and compare with the known exact value for H. (d) Discuss why values of c that are too small lead to energies that are too high. Also discuss why values of c that are too large lead to energies that are also too high! 3. The linear variational method applied to the perturbed particle in a box. For simplicity we will work in reduced units where the box size is L =1, the mass is m =1, and for convenience, ! =1 (these are atomic units). We recommend that you use the TA- supplied Python scaffold. Your objective is to develop approximate solutions to the following potential: V x ( ) = x ; x [0,1 2] V x ( ) = 1 x ( ) ; x [1 2,1] Your linear variational wavefunctions will be a superposition of N unperturbed particle-in-a-box states, where you will be able to optimize the coefficients a given N ,
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  • Summer '16
  • Alistair Sinclair

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