Please answer with an explanation

Consider a 1D harmonic oscillator potential with the Hamiltonian A E2 d2 1 H :___ _\" 2 2 0 271103332 +2 3: Now a quatic potential V is added where 17': h (a) (5 Pt) Prove that the ground state energy of the total Hamiltonian must be larger than hrs/2? (b) (5 Pt) Treat V as a perturbation, nd the ground state energy to rst order. (Hint: it is convenient to rewrite V in terms of the annihilation / creation operators for the harmonic oscillator potential.) (c) (10 Pt) Calculate the ground state energy to second order. Do you think it is a big improvement from the rst-order calculation? (d) (10 Pt) Now use the variational method, by considering a Gaussian variational ansatz sz) : e'\"52$2/2 as the ground state spatial wavefunction (1/ (3 E /(mw) is the harmonic oscillator length and a is a dimensionless variational parameter), to calculate the ground state energy. (e) (5 Pt) Which result do you think is the most accurate estimate of the true ground state energy: the rst order result in (b), the second order result in (c), or the variational result in (d)? Why