11. Consider a three-dimensional harmonic oscillator, whose state vector ) is: |) = |az)|a₂) |az) where lar), lay) and l

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11 Consider A Three Dimensional Harmonic Oscillator Whose State Vector Is Az A Az Where Lar Lay And L 1 (704.19 KiB) Viewed 45 times

11. Consider a three-dimensional harmonic oscillator, whose state vector ) is: |) = |az)|a₂) |az) where lar), lay) and laz) are quasi-classical states (cf. Complement Gy) for one- dimensional harmonic oscillators moving along Ox, Oy and Oz, respectively. Let L = Rx P be the orbital angular momentum of the three-dimensional oscillator. a. Prove: (L₂) = iħ (aza-aαv) AL₂ = √/la1² + |a₂|² and the analogous expressions for the components of L along Or and Oy. b. We now assume that: {Lz) = (Ly) =0, (L₂) = Ali> 0 Show that a must be zero. We then fix the value of A. Show that, in order to minimize ALT + ALy, we must choose: ag = -iαy = eivo V₂ (where po is an arbitrary real number). Do the expressions AL.AL, and (AL)²+ (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: |v) = cx (ar) |xnr=k, n₁=0, n₂=0) k with: (a + iat) k |Xn,=k, n=0,1 ,n₁=0, n₂ =0) |4n₂=0, n₁=0, n₂ =0) √2kk! ak Ck (α) e-la/²/2 = √k! ; ar = ¹0 √ (the results of Complement Gy and of § 4 of Complement Dyi can be used). Show that the angular dependence of Xn, k, n=0, n.-0) is (sin ei)k. L2 is 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 Lz that follows a measurement of L2 whose result was 1(1+1)ħ²? =