Coherent states of the harmonic oscillator. Among the stationary states of the harmonic oscillator (Equation 2.68) only hits the uncertainty limit ; in general, , as you found in Problem 2.12. But certain linear combinations (known as coherent 162 states) also minimize the uncertainty product. They are (as it turns out) eigenfunctions of the lowering operator:42 (the eigenvalue α can be any complex number). (a) Calculate , , , in the state . Hint: Use the technique in Example 2.5, and remember that is the hermitian conjugate of . Do not assume α is real. (b) Find and ; show that . (c) Like any other wave function, a coherent state can be expanded in terms of energy eigenstates: Show that the expansion coefficients are (d) Determine by normalizing . . (e) Now put in the time dependence: and show that remains an eigenstate of , but the eigenvalue evolves in time: So a coherent state stays coherent, and continues to minimize the uncertainty product. (f) Based on your answers to (a), (b), and (e), find and as functions of time. It helps if you write the complex number α as for real numbers C and ϕ. Comment: In a sense, coherent states behave quasi-classically. (g) Is the ground state itself a coherent state? If so, what is the eigenvalue?