11. Consider a three-dimensional harmonic oscillator, whose state vector ) is: 1V) = |az) lay) ola,) where laz), lay) an

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11 Consider A Three Dimensional Harmonic Oscillator Whose State Vector Is 1v Az Lay Ola Where Laz Lay An 1 (487.23 KiB) Viewed 26 times

11. Consider a three-dimensional harmonic oscillator, whose state vector ) is: 1V) = |az) lay) ola,) where laz), lay) and laz) are quasi-classical states (cf. Complement Gy) for one- dimensional harmonic oscillators moving along Ox, Oy and Oz, respectively. Let L = RXP be the orbital angular momentum of the three-dimensional oscillator. 801 COMPLEMENT FI a. Prove: (L2) = ifi (aza; - aja) AL; = tipla:12 +10,12 and the analogous expressions for the components of L along Ox and Oy. b. We now assume that: (L} = (Ly) = 0 , (L2) = Ali > 0 Show that a, must be zero. We then fix the value of 1. Show that, in order to minimize AL, + ALy, we must choose: Qy = -iay = (where yo is an arbitrary real number). Do the expressions AL.AL, and (AL)? + (AL)2 in this case have minimum values compatible with the inequalities obtained in question b. of the preceding exercise? c. Show that the state of a system for which the preceding conditions are satisfied is necessarily of the form: lelle) = q* C#(ar)]Xnr=k, pi=0, n. =0) with: e-lal/2 (a! + iaf)* IXn-=k, 91=0, 1,=0) len,=0, nx =0, n,=)) V2kk! ck(a) = ; VR! Qr reino va (the results of Complement Gy and of 8 4 of Complement Dvi can be used). Show that the angular dependence of Xn,=k, n2=0,n,=o) is (sin eip). Lis measured on a system in the state [). Show that the probabilities of the various possible results are given by a Poisson distribution. What results can be obtained in a measurement of L, that follows a measurement of L? whose result was l(1 + 1)??