Answer Happy

Problem 1: Consider a system described by the Hamiltonian
𝐻 = 𝐻0 + 𝐻1
Where 𝐻0 = 𝐸𝑎𝑎†𝑎 +
𝐸𝑏𝑏†𝑏 + 𝐸𝑐𝑐†𝑐 and 𝐻1 =
𝑔(𝑎†𝑏†𝑐 + 𝑐†𝑎𝑏).
The operators satisfy the commutation relations [𝑎, 𝑎†]
= [𝑏, 𝑏†] = [𝑐, 𝑐†] = 1 and all
other commutators are zero.
a) Give a complete set of commuting observables for
𝐻0.
Give a complete set of commuting observables for the total
Hamiltonian 𝐻. (20
points)
b) For 𝑔 = 0 construct the eigenstates corresponding to the four
energy
eigenvalues 𝐸𝑎, 𝐸𝑏, 𝐸𝑐 and
𝐸𝑎 + 𝐸𝑏 + 𝐸𝑐.
c) Compute the exact eigenvalues of 𝐻 for the states of the system
which reduce to
𝐸𝑐 as 𝑔 → 0. (25 points)
d) For the case of 𝐸𝑐 = 𝐸𝑎 + 𝐸𝑏,
compute the exact eigenvalue of 𝐻 which reduce to
the eigenvalue 𝐸𝑎 + 𝐸𝑏 + 𝐸𝑐 of
𝐻0 as 𝑔 → 0.
Problem 1: Consider a system described by the Hamiltonian H = H, +H Where H. Eqata + Ebbtb + Ecctc and H4 = g(atbtc +ctab). The operators satisfy the commutation relations [a, at] = [b, bt] = [c,c+] = 1 and all other commutators are zero. = = = a) Give a complete set of commuting observables for Ho. Give a complete set of commuting observables for the total Hamiltonian H. (20 points) b) For g = 0 construct the eigenstates corresponding to the four energy eigenvalues Ea, Eb, Ec and Eq + Ep + Ec. c) Compute the exact eigenvalues of H for the states of the system which reduce to E, as g = 0.(25 points) = d) For the case of Ec = Eq + Ep, compute the exact eigenvalue of H which reduce to - the eigenvalue Ea + Ep + Ec of H, as g = 0.