In this problem, you will use the variational method to find the optimal 1s wave function for the hydrogen atom starting from the trial function Φ(r) = e^ˆ’αrwith α as the variational parameter. You will minimize

with respect to α.

a. Show that

b. Obtain the result

Using the standard integrals in the Math Supplement (Appendix A)

c. Show that

Using the standard integrals in the Math Supplement.

d. You now have the result E (α) = h²Î±² / (2m_e) €“ e²Î± / (4πε₀). Minimize this function with respect to α and obtain the optimal value of α

e. Is E(α_optimal) equal to or greater than the true energy? Why?

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