2. Consider a particle of angular momentum quantum number 6=1 and the vector n = sin cos oi+sin sin ojcos (k. where 0 €

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2 Consider A Particle Of Angular Momentum Quantum Number 6 1 And The Vector N Sin Cos Oi Sin Sin Ojcos K Where 0 1 (246.65 KiB) Viewed 11 times

2. Consider a particle of angular momentum quantum number 6=1 and the vector n = sin cos oi+sin sin ojcos (k. where 0 € [0.] and o € [0, 2] are the polar and azimuthal angles that identify n, with (i, j, k) being the unit vectors along the r, y and z directions of a three-dimensional coordinate reference frame, respectively. (a) Show that the component of the angular momentum operator L, in the direction of n is given by L= sin 0 (e¯¹ºÎ+ + e¹ºĹ_) + cos 0.Î, 2 with L₁ = θ ± iLy the raising and lowering operators, and (Lr, Ly, Lz) the components of the angular momentum operator L of the particle. [6] (b) Show that the matrix representation of Ln over the basis of common eigenstates of L² and Lz {l, m)} = {1,-1), 1,0), 1, 1)} is sin 8-10 V2 COSA pio 0 Lnh sino 0 √2 0 sin 0 elo √2 COS 6 (e) Show that the eigenvalues of L are independent of the direction n that we are considering. (d) Show that the normalised eigenstate la) corresponding to the null eigenvalue of Ln reads 6 sin (a) F e|1, 1) + cos 01,0) + sin 8 12 -e¯¹⁰|1,-1), √₂ where , m) are the simultaneous eigenstates of 1² and L. with eigenvalues ª(+1) and hm, respectively. sin 0 √2 e io