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Coherent states are the eigenstates of the harmonic oscillator annihila- tion or lowering operator: ala) = ala), with complex eigenvalues a. (a) Evaluate the inner product (ala') between two coherent states. (b) Prove the (over-)completeness relation fd²a la) (a| = π 1. Hint: introduce polar coordinates a = pe in the complex plane. (c) Demonstrate that the temporal evolution of a coherent state can essentially be absorbed into a time-dependent eigenvalue a(t). (d) Find the expectation values (x(t)) a, (p(t))a, (x² (t))a, (p²(t))a, and (H(t)) in a coherent state, and show that these are consistent with the classical oscillations. Confirm that (Ax)a (Ap)a= ħ/2. (e) Confirm the representation for time-dependent coherent states |a(t)) = e-it/2 exp [a(t) at — a* (t) a] 10), and derive the explicit expression for the associated wave function (x, t) = (x|a(t)) provided in the lecture notes.