Quantum Field Theory for Leg Spinners 18/10/10 Lecture Notes 2: Review of Quantum Mechanics Lecturer: Prakash Panangaden Scribe: Jakub Závodný This lecture will briefly review some of the basic concepts of quantum mechanics, with special emphasis on the problems arising from infinite-dimensional spaces. In introductory courses of quantum mechanics these issues are often swept under the rug to keep the mathematics simple. Also, much of the categorical language developed here in Oxford is suited only for finite-dimensional spaces. However, being on our way to Quantum Field Theory, it is important to notice the differences and the problems they cause. 1 Basic Elements of Quantum Mechanics In quantum mechanics, the state space of a physical system is represented by a Hilbert space H. Two nonzero vectors of H will represent the same physical state if they are linearly dependent. Thus the state space is really the projective space of H. This is often underemphasised and H is still called the state space. 1.1 Tensor Product A vital feature of quantum mechanics is that the combined system, consisting of the systems represented by Hilbert spaces H 1 and H 2, is represented by the tensor product H 1 H 2. Finite-dimensional Hilbert spaces with the tensor product form a compact-closed monoidal category, indeed, a -compact category. However, this fails in the infinite-dimensional case. The usual formulation of the universal property of the tensor product also fails in the infinite-dimensional case. Recall the universal property of the tensor product of two vector spaces U and V : for any vector space W and any bilinear map λ : U V W, there exists a unique linear map ˆλ : U V W such that λ = ˆλ ǫ, where ǫ : U V U V is the natural embedding (u,v) u v. Since the tensor product of two Hilbert spaces is the completion of their usual vector-space tensor product, it turns out that this universal property only holds for Hilbert-Schmidt maps λ. It may seem that considering a category of Hilbert spaces and Hilbert-Schmidt maps would solve the problem, but the identity map is not Hilbert-Schmidt, so the issue is more intricate. This issue is addressed in [1], introducing nuclearity and other concepts that may be useful in axiomatising infinite-dimensional quantum mechanics. 2-1
1.2 Time Evolution The time evolution is described by a family of unitary operators U(t) : H H, so that the state vector at time t, ψ(t), is given by ψ(t) = U(t) ψ(0). It is a fundamental law of quantum mechanics that U(t) = e iht/ for a Hermitian operator H, called the Hamiltonian, which depends on the physics of the specific system considered. The infinitesimal version of the time-evolution equation then reads and is called the Schrödinger equation. i ψ(t) = H ψ(t) t Consider the eigenstates E n of the Hamiltonian and their corresponding eigenvalues E n. The time evolution of a state initalised by E n is given by ψ(t) = e iht/ E n = e ient/ E n, so it is a persistent state, only picking up a phase as time passes. Since the time evolution is linear, if we can express an initial state ψ(0) as a linear combination of E n s, it is then easy to express the state ψ(t) at any later time t. 2 Free Particle in 1D A classic example for an infinite-dimensional quantum-mechanical system, exposing many of the issues of infinite-dimensionality, is a free particle moving on a line in a potential V (x). The state space of the particle is the Hilbert space L 2 (R), whose elements are the (equivalence classes of) square-integrable complex-valued functions on R. Setting = 1 in the following and in the rest of this lecture, the Hamiltonian of the system is H = 1 2m 2 + V (x), so the equation of motion is i ψ t = 1 2m 2 ψ + V (x)ψ. 2-2
2.1 Observables The momentum operator p is defined as P = i d dx and its eigenfunctions are wave functions e ipx, where p is the eigenvalue (value of momentum). However, note that p as defined here is not really a total operator L 2 (R) L 2 (R) and none of the functions e ipx is an element of L 2 (R). Similarly, the position operator x is defined by (x ψ )(x) = x ψ (x), but again, its eigenvalues are the Dirac delta-functions δ x, which are not functions at all, let alone elements of L 2 (R). These issues are often neglected in introductory treatments. To treat them formally, the idea of rigged Hilbert spaces is used. The idea is that we pick a nuclear subspace Ω of our Hilbert space H, which is dense in but not equal to H, and form its dual Ω. Since H is isomorphic to H, we get Ω H Ω. The needed eigenfunctions of our operators will now reside in Ω, and in fact enough spectral theory can be mimicked in this context to allow for the development of quantum-mechanical constructions. More details can be found in the much recommended book [2]. 2.2 Quantisation Where did we get the Hamiltonian for the free particle? As said in the previous lecture, there is a (heuristic) process called quantisation which tries to construct a quantum theory from a given classical theory, in particular, for each classical observable we need to produce a corresponding quantal observable. Classical Quantal State space Manifold (Projective) Hilbert space Observable Real-valued function quantisation Hermitian operator Time evolution Hamiltonian Hamiltonian In our simple case, we just replace the quantities x and p in the classical Hamiltonian by the quantum mechanical operators x and p. Thus from the classical Hamiltonian H = p 2 2m + V (x) we obtain the quantum mechanical Hamiltonian H = P2 2m + V (x) = 1 2m 2-3 d 2 dx2ψ + V (x),
which is correct up to setting = 1. However, note that the operators x and p do not commute, in fact, they satisfy the canonical commutation relations 1 [x,p] = ii. Thus the above prescription is sometimes ambiguous, i.e. classically we have x p = xpx = px 2, while for quantal observables we have x 2 p xpx px 2. In such cases we just have to guess the correct expression. 3 Symmetry in Quantum Mechanics Let G be a Lie group acting on a Hilbert space H, let G 0 be a one-parameter subgroup of G, and let U(θ) be a representation of G 0. Theorem (Stone): There exists a Hermitian operator A such that U(θ) = e iθa. Moreover, if the group action commutes with the Hamiltonian H, it follows that A is conserved during time evolution. This is a quantum analogue of Nöther s theorem from classical mechanics. 4 Harmonic Oscillator In a quadratic potential V (x) = 1 2 kx2, we get a Hamiltonian H = p2 2m + 1 2 kx2. The solution of the corresponding classical system is x = Asin(ωt + φ), where ω = k/m. In anticipation of analogous solutions we write the Hamiltonian as H = p2 2m + 1 2 mω2 x 2. While the eigenvectors of H can be found using Hermite polynomials and other tricks for dealing with the differential equations involved, we will present a nicer, algebraic solution. 1 Note that in the finite-dimensional case, there are no operators X, P such that [X, P] = I. Further, it is a theorem of von Neumann that the position and momentum operators are essentially the only realisation of operators such that [X, P] = i. 2-4
Introduce the operator and its adjoint so that mω 1 a = x 2 + ip 2mω mω 1 a = x 2 ip 2mω, 1 x = [a + a], 2mω mω p = i 2 [a a] and hence ( ) H = ω a a + 1 2. Defining N = a a, we have H = ω ( N + 1 2 ) and hence to find the eigenvalues of H, it is sufficient to find the eigenvalues of N. Also, it is straightforward to check that, [a,a ] = 1, [N,a] = a and [N,a ] = a. Let n be an eigenvector of N with an eigenvalue n. (Anticipating, but not supposing, that n will be an integer.) N n = n n implies that N(a n ) = (a + a N) n = (n + 1)(a n ), i.e. that a n is an eigenvector of N with eigenvalue n + 1, unless it is the zero vector. Similarly observe that N(a n ) = ( a + an) n = (n 1)(a n ), i.e. that a n is an eigenvector of N with eigenvalue n 1, unless it is the zero vector. We call a the raising operator and a the lowering operator. Note that ψ N ψ = ψ a a ψ = a ψ 2 0, so the spectrum of N is bounded below by zero. 2 By repeated application of the lowering operator a to n, we obtain eigenvectors with lower and lower eigenvalues, or the zero vector. Since the spectrum of N is bounded below, after a finite number (at most n + 1) of applications, we must have obtained a zero 2 This implies that the spectrum of H is bounded below by 1 ω, i.e. the lowest possible energy is positive. 2 2-5
vector. If m is the least integer such that a m n is nonzero, we get a(a m n ) = 0, so also N(a m n ) = (a a)(a m n ) = 0, which means that a m n is an eigenvector of N with eigenvalue zero. But we also know that a m n is an eigenvector of N with eigenvalue n m, therefore n = m is a nonnegative integer. Furthermore, if n = 1, then a n 2 = n aa n = n (1 + a a) n = n (1 + N) n = (n + 1), so in particular, we never get a zero vector when applying a to an eigenvector of N. This means that by repeated application of the raising operator we obtain eigenvalues with arbitrarily large eigenvalues. Therefore, all nonnegative integers are eigenvalues of N. It is left as an exercise to the reader to show that all the eigenspaces are one-dimensional, so up to physically unimportant phase, there is a unique normalised n-eigenvector n for each nonnegative integer n. From this and the calculation above it then follows that a n = n + 1 n + 1 and similarly a n = n n 1. We have found all the eigenvalues of N and hence of H. Notice however that it takes more work to establish that there are no elements in the spectrum of N or H other than their respective eigenvalues. References [1] S. Abramsky, R. Blute and P. Panangaden, Nuclear and trace ideals in tensor-* categories, Journal of Pure and Applied Algebra (1999). [2] N. N. Bogoliubov, A. A. Logunov and I. T. Todorov (1975): Introduction to Axiomatic Quantum Field Theory. Reading, Mass.: W. A. Benjamin, Advanced Book Program. 2-6