Lecture 19 (Nov. 15, 2017)

Transcription

1 Lecture Quantum Theory I, Fall Lecture 19 Nov. 15, 017) 19.1 Rotations Recall that rotations are transformations of the form x i R ij x j using Einstein summation notation), where R is an orthogonal matrix, R T R = 1. This R is called a rotation matrix. For now, we will restrict to rotations R with det R = +1 orientation-preserving or proper rotations). Every rotation R of space corresponds to a unitary operator DR) on the Hilbert space, which satisfies DR 1 )DR ) = DR 1 R ). 19.1) We will discuss this composition property more later. A quantum state transforms as α α R under this rotation, such that α R = DR) α. 19.) For a vector operator V i, with i = 1,..., d, we require Expanding the transformed bra and ket, this is β R V i α R = R ij β V j α. 19.3) β D R)V i DR) α = R ij β V j α. 19.4) This is true for any states α, β, which implies that the operators must be equal: D R)V i DR) = R ij V j 19.5) holds as an operator equation. Consider the infinitesimal rotation R = 1 ω. The orthogonality condition, R T R = 1, then implies that ω T = ω. Thus, ω is a real, antisymmetric matrix. We can then expand DR) in the form i DR) 1 ωijj ij + O ω ). 19.6) ij This expansion identifies the objects J ij = J ji as the Hermitian generators of rotations. Note that the antisymmetry of J ij follows from the antisymmetry of ω ij. Let us now specialize to three dimensions, d = 3. In this case, we can write i DR) = 1 J ω + J ω + J ω ) + O ω ) , 19.7) where we have used the antisymmetry of J ij to group terms. We define J 1 := J 3, J := J 31, J 3 := J 1, 19.8) 1 J i = ɛ ijk J jk, 19.9) where ɛ ijk is the totally antisymmetric symbol with ɛ 13 = +1, known as the Levi Civita symbol. We can similarly define θ 1 := ω 3, θ := ω 31, θ 3 := ω 1, 19.10) 1 θ i = ɛ ijk ω jk, ω ij = ɛ ijk θ k )

2 Lecture Quantum Theory I, Fall Then, we have Thus, Note that i DR) = 1 θ k J k + O θ ). 19.1) R ij = δ ij ɛ ijk θk + O θ ) ) x i x i = R ij x j = δ ij ɛ ijk θ k )x j = x i ɛ ijk x j θ k, 19.14) x x = x + θ x ) Thus, the meaning of θ is that x is rotated by an angle θ about the θ-direction. We now define J k to be the components of the angular momentum. First, we will derive the commutation relations of J k with any vector operator V i. We start with the equation D R)V i DR) = R ij V j 19.16) and take R = 1 ω with ω infinitesimal. The left-hand side then becomes ) iθ k J k 1 + V i 1 iθ ) lj l = V i + iθk [Jk, V i ], 19.17) while the right-hand side becomes Thus, we conclude that R ij V j = V i ɛ ijk V j θ k ) [J k, V i ] = iɛ kij V j ) We can then use a combination of rotations to deduce the angular momentum algebra, via DR 1 )DR ) = DR 1 R ). 19.0) In particular, this composition rule implies that ) ) 1 1 DRφ)DR θ )D R φ = D R φ R θ R φ. 19.1) The rotation R φ R θ R 1 can be written as a single rotation R θ φ for some θ. As θ itself is a vector, for φ infinitesimal, we have θ = θ + φ θ. 19.) If we take θ to be infinitesimal, we have so 19.1) becomes iθ DR θ ) 1 k J k = + O θ ), 19.3) ) θ k DR φ )J k D R 1 φ = θk J k 19.4) The left-hand side of this equation, for infinitesimal φ is iφ j θ 1 j J k )J k 1 + iφ ) lj l = θ k J k iθ kφ j [J j, J k ] +, 19.5)

3 Lecture Quantum Theory I, Fall while the right-hand side is which leads us to conclude that θ k J k + ɛ jkl φ j θ k J l +, 19.6) [J i, J j ] = iɛ ijk J k. 19.7) This is the angular momentum commutation algebra. Note that this matches the commutation relation for a vector operator with the angular momentum operator, so this shows that the angular momentum operator is a vector. In general, we can write the angular momentum as J = L + S, 19.8) with L = x p is the orbital angular momentum and S is an internal property that commutes with x, p, etc. We can check that L on its own satisfies the angular momentum commutation algebra, so the operator J will satisfy the angular momentum algebra if S does. The operator S is the spin operator. If the Hamiltonian is rotationally invariant, then [J i, H] = 0, which implies that and so angular momentum is conserved. dj i = 0, 19.9) dt Eigensystem of Angular Momentum Let us now understand the implications of the commutation algebra [J i, J j ] = iɛ ijk J k ) You will show on the homework that [ J ], J i = ) This means that we can diagonalize J and one component of the angular momentum, say J z, simultaneously. We can then label the eigenstates of J z by j, m, with J j, m = a j, m, J z j, m = b j, m, 19.3) for some eigenvalues a, b. The meanings of the values j and m will become apparent shortly. It is useful to define the ladder operators which satisfy [J +, J ] = J z, [J z, J ] = ±J, Using these commutation relations, we see that J ± = Jx ± ij y, 19.33) ± ± ± [ J, J ] = ) J z J j, m ± ) = J J ± z ± J ) ± j, m = b ± )J ± j, m ) ) Thus, J± j, m is also an eigenstates of Jz with eigenvalue b ±.

4 Lecture Quantum Theory I, Fall We can write J = Jx + Jy + Jz = J 1 z + J + J + J J + ) = Jz + 1 ) J + J + + J J, 19.36) which tells us that J Jz is positive semi-definite, j, m J Jz j, m ) This implies that a b 0 for all eigenstates. For a fixed a, this means that b has a maximum value b max. This seems to be in conflict with the statement that we can use J± to raise or lower the eigenvalue arbitrarily. We conclude that at b = +b max the state must be annihilated by J +, and similarly, at b = b max the state must be annihilated by J. Call max the state with b = +b max. Then, we have which implies J + max = 0, 19.38) J J + max = ) Expanding the ladder operators, this becomes J x ij y )J x + ij y ) max = J Jz ) J z max = ) This gives us a b max b max = 0, 19.41) a = b max b max + ). 19.4) Repeating this argument for the state min with b = b max yields the same result. We now note that, because J + increases the eigenvalue b, and this eigenvalue is bounded above by b max, we must be able to reach max from min by repeatedly applying J +. Say we can reach max from min by n applications of J +. This implies that b max = b max + n, 19.43) so n b max = = j, 19.44) with j 1 Z. We can then read off the eigenvalues for the state j, m, a = jj + 1), b = m, 19.45) and see that m can take any of the j + 1 values j, j + 1,..., j 1, j.

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