Quantum Theory I, Lecture 20 Notes

Lecture 20 8.321 Quantum Theory I, Fall 2017 Lecture 20 (Nov. 20, 2017) 20.1 Matrix Elements of Angular Momentum Operators Assume that we have normalized |j, mi. By definition, since these are eigenstates of J 2 and Jz , these operators are diagonal, with hj 0 , m0 |J 2 |j, mi = j(j + 1)~2 δjj 0 δmm0 , hj 0 , m0 |Jz |j, mi = m~δjj 0 δmm0 . (20.1) Now we only need to compute the matrix elements of Jx and Jy . We will make use of the identity
hj, m|J+ J− |j, mi = hj, m|J 2 − Jz2 − ~Jz |jmi = ~2 j(j + 1) − m2 − m . (20.2) We also know that (+) J+ |j, mi = cj,m |j, m + 1i . (20.3) † We still need to determine the coefficient cj,m . Using (20.2) and J− = J+ , we see that (+) (+) 2 cj,m = ~2 (j(j + 1) − m(m + 1)) = ~2 (j − m)(j + m + 1) . (20.4) (+) We will choose (by convention) for cj,m to be real and positive, which then gives us p J+ |j, mi = ~ (j − m)(j + m + 1)|j, m + 1i . (20.5) A similar argument gives us p J− |j, mi = ~ (j + m)(j − m + 1)|j, m − 1i . (20.6) Packaging these results together, we have p hj 0 , m0 |J± |j, mi = ~ (j ∓ m)(j ± m + 1)δjj 0 δm0 ,m±1 . (20.7) We see that when m = +j, the state is annihilated by J+ , and when m = −j, the state is annihilated by J− , exactly as required. We can now easily calculate the matrix elements of Jx and Jy , as they are linear combinations of J± . This allows us to explicitly write matrix representations of J for fixed j. As an example, consider j = 1. In this case,   1 0 0 Jz = ~0 0 0  , (20.8) 0 0 −1 where the basis is ordered as m = +1, 0, −1. Using the matrix elements for J± , we can write down √   0 2 0 √ J+ + J− ~ √ Jx = =  2 √0 2 , 2 2 0 2 0 (20.9) √   0 − 2i 0 √ Jx − Jy ~ √ Jy = = 2i √0 − 2i . 2i 2 0 2i 0 Lecture 20 8.321 Quantum Theory I, Fall 2017 For any fixed j, we can now also construct matrix elements of the finite rotation operator D(R). For a fixed j, the angular momentum operators are (2j + 1) × (2j + 1) Hermitian matrices, and so the rotation matrix D(R) will be (2j + 1) × (2j + 1) unitary matrices. Consider a rotation by angle θ about the axis n̂. First, we construct an infinitesimal rotation by δθ about the axis n̂:
i D(Rδθ ) = 1 − (J · n̂)δθ + O δθ2 . ~ (20.10) We can then build up a finite rotation by multiplying infinitesimal rotations N times and taking N → ∞, D(R(θ, n̂)) = lim (D(θ/N, n̂))N N →∞  
N i θ 2 (20.11) = lim 1 − (J · n̂) + O (θ/N ) N →∞ N ~ = e−iθJ·n̂/~ . 20.2 Rotation Groups In general, symmetry operations form a mathematical structure called a group. Definition 1. A set {gi } := G forms a group if there exists a multiplication operation · such that: • for all g1 , g2 ∈ G, g1 · g2 ∈ G; • multiplication is associative: for all g1 , g2 , g3 ∈ G, (g1 · g2 ) · g3 = g1 · (g2 · g3 ) ; (20.12) 1·g = g·1 = g; (20.13) • there exists 1 ∈ G such that • for every g ∈ G, there exists g −1 ∈ G such that g · g −1 = g −1 · g = 1 . (20.14) For example, the set of 3 × 3 orthogonal matrices forms a group under the operation of matrix multiplication. This group is known as the orthogonal group O(3). If we restrict our attention to rotations with det R = +1, then we get a subgroup (a subset that is itself also a group under matrix multiplication) called the special orthogonal group SO(3). There is another pertinent group to the discussion of rotations, which is the group of 2 × 2 unitary matrices satisfying det U = 1. Any matrix of this kind can be written in the form   a b U (a, b) = , (20.15) −b∗ a∗ with |a|2 + |b|2 = 1. Such matrices form a group under matrix multiplication, known as the special unitary group SU(2). In quantum mechanics, rotations are represented by unitary operators D(R). These unitary operators must themselves form a group, because the product of two of these operators must give
−1 another, the composition is associative, there is an identity operation, and D R is the inverse of D(R). In particular, we can see that the D(R) must satisfy the same group identities as the group of rotations R. Lecture 20 8.321 Quantum Theory I, Fall 2017 The matrix elements of D(R) are (j) Dm0 ,m (R) = hj, m0 |e−iθJ·n̂/~ |j, mi . (20.16) Note that we have chosen the same j for both the bra and ket; this is because D(R) commutes with J 2 , so a rotation cannot change the value of J 2 . As a result, in the full ket space, the (infinitedimensional) matrix for D(R) is block diagonal in the (j, m) basis, where each block corresponds to one value of j. The block corresponding to j will be a (2j + 1) × (2j + 1) submatrix. An explicit set of matrices that satisfies the same group identities as a group is called a (linear) group representation. Thus, the D(R) form a representation of SO(3). In particular, this representation is called a completely reducible representation, because it can be written in block diagonal (j) form. Each block on the diagonal of D(R), taken as a standalone matrix Dm0 ,m for fixed j, is an irreducible representation (sometimes referred to as an irrep), as it has no invariant subspaces. Consider j = 1/2. In this case, m and m0 take on the values ± 21 . The rotation operator matrix elements are (1/2) Dm0 ,m (θ, n̂) = hm0 |e−iθσ·n̂/2 |mi . (20.17) That is, the rotation operators are D(1/2) (θ, n̂) = e−iθσ·n̂/2 θ θ = cos − i(σ · n̂) sin . 2 2 (20.18) Explicitly, D (1/2)  (θ, n̂) = cos 2θ − inz sin 2θ (−inx + ny ) sin 2θ (−inx − ny ) sin 2θ cos 2θ + inz sin 2θ  . (20.19) This may seem unusual, because this is an SU(2) matrix. The operators D(1/2) (θ, n̂) form a representation of SU(2), even though we originally set out to find a representation of SO(3). Rotations of 3D space can be characterized either by an SO(3) matrix R (a three-dimensional irrep) or by an SU(2) matrix U (a two-dimensional irrep). But SO(3) and SU(2) are not in one-toone correspondence. Suppose we consider a rotation by θ = 2π along some axis n̂. For SO(3), while for SU(2), R(2π, n̂) = 13 , (20.20) D(1/2) (2π, n̂) = e−iπσ·n̂ = −12 . (20.21) The result is that for any spin- 12 system, the phase of the wavefunction is rotated by π under a physical 2π rotation.