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Some Linear Algebra and Elementary Quantum Mechanics

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Some Linear Algebra and Elementary Quantum Mechanics Representation theory and quantum mechanics tutorial Some linear algebra and elementary quantum mechanics Justin Campbell July 25, 2017 1 Hilbert spaces 1.1 Recall that a real inner product space is a vector space V over R equipped with a bilinear form h ; i : V × V −! R satisfying (i) (symmetric) we have hv; wi = hw; vi for all v; w 2 V , (ii) (positive-definite) we have hv; vi > 0 for all v 2 V such that v 6= 0. In a real inner product space V one can make sense of the length of a vector v 2 V : jvj := phv; vi; as well as the angle θ between two vectors v; w 2 V , using the formula hv; wi cos θ = : jvjjwj In particular, two vectors v; w 2 V are orthogonal if and only if hv; wi = 0. For example, the Euclidean space Rn with the dot product is an inner product space. The Gram-Schmidt algorithm shows that any n-dimensional real inner product space V admits an orthonormal basis, meaning a basis consisting of pairwise orthogonal vectors of length one. The resulting isomorphism V !~ Rn sends the inner product in V to the dot product in Rn. But many interesting inner product spaces are infinite- dimensional. Exercise 1.1.1. Let C1(S1) be the space of smooth R-valued functions on the circle, which we can view as smooth functions f : R ! R which are periodic of period, say, 2π. Show that C1(S1) is an inner product space under the pairing Z Z 2π hf; gi := f(θ)g(θ) dθ = f(θ)g(θ) dθ: S1 0 If we work with vector spaces over the complex numbers C, as we must to do quantum mechanics, the positive-definiteness condition forces us to abandon bilinear forms. Instead we use Hermitian, i.e. conjugate- symmetric, forms. Definition 1.1.2. A Hilbert space or complete complex inner product space is a vector space H over C equipped with a form h ; i : H × H −! C satisfying (i) the map v 7! hv; wi is linear for any w 2 H , 1 (ii) (Hermitian) we have hv; wi = hw; vi for all v; w 2 H , (iii) (positive-definite) we have hv; vi > 0 for all v 2 H such that v 6= 0. (iv) (complete) any Cauchy sequence in H converges. Here the notions of Cauchy and convergent sequence are defined using the norm jvj := phv; vi; which in turn gives rise to a metric. The \angle" θ between two vectors v; w in any complex inner product space can be defined so that Re hv; wi cos θ = : jvjjwj In particular, they are orthogonal if and only if hv; wi = 0. Remark 1.1.3. The axiom of completeness is automatically satisfied in the finite-dimensional case. Any complex inner product space V injects into a canonical Hilbert space V^ , called its completion. Exercise 1.1.4. Construct the completion V^ ⊃ V of an arbitrary complex inner product space V , including its unique Hermitian form extending the one on V (hint: imitate the construction of the real numbers using Cauchy sequences). Show that for any Hilbert space H which contains V as a dense subspace with the induced inner product, there is a unique isomorphism V^ !~ H which preserves the Hermitian forms. The basic example of a Hilbert space is the standard n-dimensional complex vector space Cn with the pairing X h(ai); (bi)i := aibi; i which is the Hermitian version of the dot product. As in the real case, the Gram-Schmidt algorithm shows that any finite-dimensional Hilbert space admits an orthonormal basis. One can define an infinite-dimensional complex inner product space C1(S1; C) analogously to Exercise 1.1.1: a vector is a periodic function f : R ! C of period 2π, and the inner product is defined by Z Z 2π hf; gi := f(θ)g(θ) dθ = f(θ)g(θ) dθ: S1 0 This space is not complete; its completion is the Hilbert space denoted by L2(S1). 1.2 Let H1 and H2 be Hilbert spaces and A : H1 ! H2 a C-linear map. ∗ Definition 1.2.1. A (Hermitian) adjoint of A is a linear map A : H2 ! H1 satisfying ∗ hAv; wiH2 = hv; A wiH1 for all v 2 H1, w 2 H2. If H1 and H2 are finite-dimensional, then any map A : H1 ! H2 admits a unique adjoint. Namely, n m one can choose orthonormal bases to identify H1 and H2 with C and C respectively, endowed with their Hermitian dot products. Then A is realized as a matrix, and its adjoint A∗ = A> is simply the conjugate transpose of A. This observation generalizes as follows. Theorem 1.2.2. A continuous linear map between Hilbert spaces admits a unique adjoint, which is then also continuous. One proves this by reducing to the case of a continuous linear functional λ : H ! C. Then the Riesz representation theorem guarantees that there exists a vector v 2 H such that λ(w) = hw; vi 2 for all w 2 H . A continuous linear operator A : H ! H is called self-adjoint or Hermitian if A = A∗, i.e. we have hAv; wi = hv; Awi for all v; w 2 H . For example, let H = C with the pairing hz; wi = zw.A C-linear operator on C is multiplication by some z 2 C, and z∗ = z. Thus z is self-adjoint if and only if z = z, i.e. z 2 R. Remark 1.2.3. Many subtleties can occur in the case that H is infinite-dimensional, where one sometimes needs to consider discontinuous or partially-defined linear operators. Then this definition gives too large a class of self-adjoint operators, and one must also impose the condition that the domain of A∗ is equal to the domain of A. The key property of self-adjoint operators is expressed in the following theorem. Theorem 1.2.4 (Spectral theorem for self-adjoint operators). If A : H ! H is a self-adjoint operator, then the eigenvalues of A are real and H admits an orthonormal basis consisting of eigenvectors for A. Proof. If v 2 H is an eigenvector of A with eigenvalue λ, then we have λhv; vi = hAv; vi = hv; Avi = λhv; vi: Thus λ = λ, i.e. λ 2 R. We give the rest of the proof under the assumption that H is finite-dimensional. The infinite-dimensional case is much deeper, and more subtle: for example, the definition of \basis" has to be modified. The theorem being obvious in the case that H = 0, we may proceed by induction, assuming that the theorem holds when dim H = d − 1. Suppose dim H = d and let v 2 H be an eigenvector of A, which must exist because H is a finite-dimensional complex vector space. We can normalize so that jvj = 1. Let W := fw 2 H j hw; vi = 0g be the subspace orthogonal to v. Then for any w 2 W , we have hAw; vi = hw; A∗vi = hw; Avi = λhw; vi = 0; meaning A preserves the subspace W . Since dim W = d − 1, the inductive hypothesis implies that W admits an orthonormal basis consisting of eigenvectors for A. Adding v to this basis completes the proof. 1.3 A continuous linear operator A : H ! H is called unitary if A∗ = A−1, or equivalently A is invertible and preserves the Hermitian inner product, i.e. hAv; Awi = hv; wi for v; w 2 H . Exercise 1.3.1. Consider the Hilbert space Cn with the Hermitian dot product. Show that a linear map A : Cn ! Cn, i.e. an n × n complex matrix, is unitary if and only the columns of A form an orthonormal basis of Cn. More generally, a linear operator A : H ! H on an n-dimensional Hilbert space H is unitary if and only if for any orthonormal basis v1; ··· ; vn of H , the vectors Av1; ··· ; Avn form an orthonormal basis. Theorem 1.2.4 can be reformulated as follows for matrices: if A 2 Matn×n(C) is self-adjoint, then there exists an n × n unitary matrix U such that UAU −1 is a diagonal matrix with real entries. The n × n unitary matrices form a group ∗ ∗ Un := fA 2 GLn(C) j A A = AA = Ig 3 called the nth unitary group. This is a real algebraic group, since Un ⊂ GLn(C) ⊂ GL2n(R): One can also consider the nth special unitary group SUn := Un \ SLn(C) = fA 2 Un j det A = 1g: The first unitary group 1 U1 = S = fz 2 C j jzj = 1g is the circle group. Moreover, SU1 = 1 is the trivial group. Exercise 1.3.2. Prove that jdet Aj = 1 for a unitary matrix A 2 Un, and that the image of det × Un ⊂ GLn(C) −! C is U1. 1.4 Similarly to the case of the orthogonal group, one computes that the Lie algebra ∗ un := fA 2 gln(C) j A = −Ag; of Un consists of skew-Hermitian matrices. Exercise 1.4.1. Show that a skew-Hermitian matrix A 2 un has purely imaginary diagonal entries, and in particular that tr A is purely imaginary. Similarly, the Lie algebra of SUn is sun := un \ sln(C) = fA 2 un j tr A = 0g: Let H be a Hilbert space. Observe that A 7! iA defines a bijection fself-adjoint operators on H g−!f~ skew-Hermitian operators on H g with inverse A 7! −iA.
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