Consider a mass moving in a spherically symmetric potential

Question

Problem - Consider a mass moving in a spherically symmetric potential V(r) = kr, k > 0. Suppose you guess that the ground state (l=0) has a radial wave function, R_1,0 = √(1/2a³)e^-(r/2a)

1) Let a_min be the value of the constant a that minimizes the expectation of Hamiltonian. Find an expression for a_min in terms of h, k and m. Use ₀∫^infinity ze^-z dz = 1.

2) Explain physically why a_min decreases as k increases.

3) This guess for the found state wave function does not fall off fast enough as r goes to infinity. What is the correct asymptotic behavior for very large r?