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PHYS 811: Assignment 8 1. Consider the operator J deﬁned only by the commutation relations J × J = i~J . Using the basic properties of this operator proved in class, show that: (i) ⟨Jx ⟩ = ⟨Jy ⟩ = 0 in a state |jm⟩ that is an eigenstate of J 2 and Jz . ] 2 [ (ii) Show that in these states ⟨Jx2 ⟩ = ⟨Jy2 ⟩ = ~2 j(j + 1) − m2 (use symmetry arguments to relate ⟨Jx2 ⟩ to ⟨Jy2 ⟩). (iii) Check that ∆Jx ∆Jy from part (ii) satisﬁes the inequality imposed by the uncertainty principle [Eq. (9.2.9)]. (iv) Show that the uncertainty bound is saturated in the state |j, ±j⟩. (j) 2. Denote by Ji the jth block of the matrix of the operator Ji , i = x, y, or z, in the eigenbasis |jm⟩ of the angular momentum operators J2 and Jz . (j) (i) Argue that the eigenvalues of Jx (j) (j) and Jy are the same as those of Jz , namely, (j) ~j, ~(j − 1), . . . , (−~j). Generalize the result to J ≡ θ̂ · ⃗J(j) . θ̂ (ii) Show that K (j) θ̂ = [J (j) θ̂ = Jz , what happens when Kz acts on a column vector ejm of Hint: In the case J (j) θ̂ − j~][J (j) θ̂ − (j − 1)~] · · · [J (j) (j) θ̂ + j~] = 0. (j) length j with elements (ejm )i = δim ? What about an arbitrary superpositions of such vectors? ( ) ( ) ( ) (j) 2j+1 (j) 0 (j) 1 (iii) Show that (ii) implies that J is a linear combination of J , J , ..., θ̂ θ̂ θ̂ ( )2j+k ( )2j (j) (j) , k = 2, 3, . . . . . Argue that the same goes for J J θ̂ θ̂ (iv) Show that (iii) implies that the inﬁnite sum in the deﬁnition of the matrices D(j) [Rθ ] of the rotation operator U [Rθ ] D(j) [Rθ ] = exp(−iθ · ⃗J(j) /~) = ( ) ∞ ∑ 1 i n ( ⃗ (j) )n − θ·J n! ~ n=0 can be reduced to a ﬁnite sum. 3. Consider the rotation by an angle θ around vector θ̂. Let U [Rθ ] be the operator corresponding to this rotation and acting on vectors in the Hilbert space. Denote the jth block of the matrix of this operator in the eigenbasis |jm⟩ of the angular momentum operators J2 and Jz by D(j) [Rθ ]. Assume that the relation between the matrix D(j) and the respective matrices (j) Ji , i = x, y, or z, of the components of the angular momentum operator is given by D(j) [Rθ ] = exp(−iθ · ⃗J(j) /~) (j) (j) (j) where θ · ⃗J(j) = θx Jx + θy Jy + θz Jz . Show that (i) D(1/2) [Rθ ] = cos(θ/2)I(1/2) − 2i sin(θ/2)θ̂ · ⃗J(1/2) ~ where I(j) is the unit matrix of dimension 2j + 1. (ii) ( (1) D [Rθx x̂ ] = (cos θx − 1) (1) Jx ~ Hint: Use the following relation for the matrices J [J (j) θ̂ (j) θ̂ − j~][J (j) θ̂ )2 (1) − i sin θx Jx + I(1) ~ ≡ θ̂ · ⃗J(j) − (j − 1)~] · · · [J (j) θ̂ + j~] = 0. 4. Consider the family of states |jj⟩, . . . , |jm⟩, . . . , |j, −j⟩. One refers to them as states of the same magnitude but diﬀerent orientation of angular momentum. If one takes this remark literally, i.e., in the classical sense, one is led to believe that one may rotate these into each other, as is the case for classical states with these properties. Consider, for instance, the family |11⟩, |1, 0⟩, |1, −1⟩. It may seem, for example, that the state with zero angular momentum along the z axis, |1, 0⟩, may be obtained by rotating |11⟩ by some (π/2?) angle about the x axis. (i) Using the rotation matrix D(1) [Rθx x̂ ] show that ) ( (1) (1) 2 J J x x − i sin θx + I(1) |11⟩ |10⟩ ̸= D(1) [Rθx x̂ ]|11⟩ = (cos θx − 1) ~ ~ for any θx , where here |10⟩ = [0, 1, 0]T and |11⟩ = [1, 0, 0]T . The error stems from the fact that classical reasoning should be applied to ⟨J ⟩, which responds to rotations like an ordinary vector, and not directly to |jm⟩, which corresponds a vector in Hilbert space. (ii) Verify that ⟨J ⟩ responds to rotations like its classical counterpart, by showing that ⟨J ⟩ in the state D(1) [Rθx x̂ ]|11⟩ is ~[− sin(θx )ŷ + cos(θx )ẑ]. (iii) Give arguments showing why we cannot always satisfy |jm′ ⟩ = D(j) [Rθx x̂ ]|jm⟩ (where |jm⟩ are still proper column vectors of size 2j +1) or more generally, for two normalized column vectors |ψj′ ⟩ and |ψj ⟩, to satisfy |ψj′ ⟩ = D(j) [Rθx x̂ ]|ψj ⟩ by any choice of R. Of course, one can ﬁnd a unitary matrix Vj that will perform the required transformations, but it will not be a rotation matrix corresponding to U [Rθ ].