4. Use the variational method with wave functions of the formφα(r)=raπ21r2+a2andφα(re−r/a√πa3to estimate the ground state energy of a particle subject to a Coulomb potentialV(r−e2/r.Note thatthe second wave function above reduces, for the correct value ofa,to the exact ground state wave functionfor the Coulomb problem.5. Evaluate tofrst order the energy shifts in the spectrum of the harmonic oscillator HamiltonianH0=12~ω(ˆq2+ˆp2) due to a perturbationˆV=λˆq4.(Hereˆqand ˆpare dimensionless position and momentumvariables, such that [ˆq,ˆp]=i. See Chapter 3 of the class notes for details, and for useful techniques forcalculating expectation values of powers of ˆq=(a+a+)/√2.)6. A system with HamiltonianH(0)is subject to a perturbationH(1),which in a certain ONB can be repre-sented by the following matriceshH(0)i=⎛⎜⎜⎜⎜⎝ε0000004ε00006ε008ε00010ε0⎞⎟⎟⎟⎟⎠hH(1)i=⎛⎜⎜⎜⎜⎝γ∆0
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