Discuss an isotropic harmonic oscillator

Question

Discussion:

Q: Consider a 3-dimensional, spherically symmetric, isotropic harmonic oscillator with a potential energy of

The Hamiltonian in this case is:

a. Use the trial function φ_a= e^-arand ?nd the value of the parameter a that the energy, and ?nd that minimum energy.

b. Repeat the process for the trial function  φ_b= e^-br2 and ?nd the parameter b and the minimum energy.

c. The actual minimum energy turns out to be E_o=3ωh/2(i.e. each dimension contributes the usual hω/2. Compare the values you calculated in parts a and b to this value. Which is closer? Why do you think that is the case?