Question d) e) f) 3. Griffiths 3.35***Problem 3.35 Coherent states of the harmonic oscillator. Among the stationarystates of the harmonic oscillator (\n) = Yn(x), Equation 2.67) only n= 0 hits theuncertainty limit (ơx0p = ħ/2); in general, ơxop = (2n + 1)h/2, as you foundin Problem 2.12. But certain linear combinations (known as coherent states) alsominimize the uncertainty product. They are (as it turns out) eigenfunctions of thelowering operator:32a_la) = a|a)(the eigenvalue a can be any complex number).(a) Calculate (x), (x²), (p), (p²) in the state |æ). Hint: Use the technique inExample 2.5, and remember that a4 is the hermitian conjugate of a-. Do notassume a is real.(b) Find ox andOpishow thatOxOp= h/2.(c) Like any other wave function, a coherent state can be expanded in terms ofenergy eigenstates:00la) = Gulm).%3Dn=0Show that the expansion coefficients area"Co.Jn!(d) Determine co by normalizing |æ). Answer: exp(-|a|²/2).(e) Now put in the time dependence:In) → e-i Ent/h \n),and show that |(t)) remains an eigenstate of a-, but the eigenvalue evolvesin time:a (1) = e¬iw!«.So a coherent state stays coherent, and continues to minimize the uncertaintyproduct.(f) Is the ground state (|n = 0)) itself a coherent state? If so, what is the eigen-value?

It is given that, energy eigen state of a coherent state is, cn=αnn!c0 and |α>=∑cn|n>n=0∞…

(c) Like any other wave function, a coherent state can be expanded in terms of energy eigenstates: la) = [c₁|n). 11=0 Show that the expansion coefficients are a" √n! Cn Co- (d) Determine co by normalizing la). Answer: exp(-la|²/2). (e) Now put in the time dependence: In) → e-i Ent/hIn). and show that la(1)) remains an eigenstate of a_, but the eigenvalue evolves in time: a(t) = e-ita. α. So a coherent state stays coherent, and continues to minimize the uncertainty product. (f) Is the ground state (n = 0)) itself a coherent state? If so, what is the eigen- value?

(c) Like any other wave function, a coherent state can be expanded in terms of energy eigenstates: la) = [c₁|n). 11=0 Show that the expansion coefficients are a" √n! Cn Co- (d) Determine co by normalizing la). Answer: exp(-la|²/2). (e) Now put in the time dependence: In) → e-i Ent/hIn). and show that la(1)) remains an eigenstate of a_, but the eigenvalue evolves in time: a(t) = e-ita. α. So a coherent state stays coherent, and continues to minimize the uncertainty product. (f) Is the ground state (n = 0)) itself a coherent state? If so, what is the eigen- value?