Fortschritte der Physik Fortschr. Phys. 63, No. 9–10, 644–658 (2015) / DOI 10.1002/prop.201500023 Progress Review Paper of Physics Functional analysis and quantum mechanics: an introduction for physicists Kedar S. Ranade∗ Received 4 May 2015, revised 4 May 2015, accepted 8 June 2015 Published online 20 August 2015 It is the intention of this tutorial to introduce basic We give an introduction to certain topics from functional anal- concepts of functional analysis to physicists and to point ysis which are relevant for physics in general and in particular out their influence on quantum mechanics. We aim at for quantum mechanics. Starting from some examples, we dis- mathematical rigour in terminology, but we will leave out cuss the theory of Hilbert spaces, spectral theory of unbounded proofs. Our approach is not necessarily consistent with a operators, distributions and their applications and present mathematics textbook building up from bottom to top by proving every theorem from some basic axioms. Rather, some facts from operator algebras. We do not give proofs, but we strive at an intuitive understanding from concepts present examples and analogies from physics which should be and analogies known either from basic linear algebra or useful to get a feeling for the topics considered. used in basic quantum mechanics. Prerequisites for reading this tutorial are a knowledge of textbook quantum mechanics (such as wave mechan- ics and Dirac notation) including some basic quantum 1 Introduction optics (harmonic oscillator, ladder operators, coherent states etc.), but excluding quantum field theory; from It is well-known that physics and mathematics are mathematics, knowledge of analysis and linear algebra is closely interconnected sciences. New results in physics required.1 lead to new branches of mathematics, and new concepts from mathematics are used to describe physical situa- tions and to predict new physical results. In the study 2 Hilbert space theory of physics quantum mechanics plays a fundamental role for almost all fields of modern physics: from quan- The formalism of quantum mechanics uses the concept tum optics, quantum information to solid-state physics, of a Hilbert space. In this section we will elaborate on the high-energy physics, particle physics etc. A similar role theory and give a full classification of Hilbert spaces, i. e. is enjoyed in mathematics by functional analysis in the we list (in some sense) all possible Hilbert spaces. The theory of differential equations, numerical methods and mathematics covered here can be found in several text- elsewhere. books on functional analysis, e. g. [1–3]. While every physicist knows quantum mechanics and every mathematician knows functional analysis, they of- ten do not know the connection between these two fields—both of which were founded at the beginning of the 20th century. While every student of physics at- 1 This manuscript is partially based on a series of lectures enti- tends lectures on higher mathematics, such as analysis of tled Mathematische Aspekte der Quantenmechanik presented one or several variables, linear algebra, differential equa- (in german) at the Institut fur¨ Quantenphysik, Universit¨at Ulm, tions and complex analysis, it is not so common that from April to August 2014. he gets to know functional analysis. Though it is possi- Institut fur¨ Quantenphysik, Universit¨at Ulm, and Center for Inte- ble to understand the basics of quantum mechanics with grated Quantum Science and Technology (IQST), Albert-Einstein- pure linear algebra, a deeper understanding is gained Allee 11, D-89081 Ulm, Deutschland, Germany ∗ by knowing at least the very structure of functional Corresponding author: E-mail: [email protected], analysis. Phone: +49/731/50-22783 644 Wiley Online Library C 2015 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim Fortschritte der Physik Fortschr. Phys. 63, No. 9–10 (2015) Progress Review Paper of Physics 2.1 Introductory examples function on [0; ∞) with a possibly complex eigenvalue α, provided that Im α>0. How is this possible? We start this tutorial by presenting four examples, which show that the na¨ıve use of linear algebra fails in certain situations.2 2.1.4 Eigenvalues of hermitian operators II In the previous example, one may argue that the radius 2.1.1 Commutators and traces cannot be negative and some part of the eigenfunction is cut off. Now we give a more dramatic example. First It is a simple exercise in linear algebra to show that we consider an abstract hermitian operator A with eigen- there holds Tr AB = Tr BA for any two matrices A and B vector v and eigenvalue λ: Av = λv. Hermiticity of A is (prove it by coordinate representation), which may be defined by Ax|y=x|Ay for all x, y ∈ H, and textbook rewritten as a trace of a commutator: Tr [A, B] = 0. In quantum mechanics (cf. e. g. Schwabl [10]) tells us that quantum mechanics, one requires, by analogy to the eigenvalues of hermitian operators are real by the follow- Poisson bracket of classical mechanics, [xˆ, pˆ] = i1H for ing simple argument: we calculate position and momentum operators xˆ and pˆ, respectively (known as canonical quantisation). Taking the trace on ∗ λ v|v=λv|v=Av|v=v|Av=v|λv=λv|v, (1) both sides of this equation yields 0 = i dim H.3 So does quantum mechanics really exist? so v|v = 0 yields λ = λ∗, and this implies λ ∈ R. Now take a spinless pointlike one-dimensional par- 2.1.2 Distributions ticle in position space, which is described by a square- integrable function on R. Consider the hermitian opera- ˆ = 3 + 3 = = ∂ Consider a particle in a box (square-well potential) with tor A xˆ pˆ pˆxˆ with the usual xˆ x and pˆ i ∂x and infinitely high walls: V (√x) = 0, if |x| ≤ a and V (x) =∞ take the function (plotted in the diagram) = 15 2 − 2 otherwise. Let (x) 4a5/2 (a x ) be the wavefunc- 1 −3/2 − 1 tion in the interior part which shall vanish outside f (x) = √ |x| e 4x2 . (2) 2 of the box. If we want to calculate the variance of the energy Hˆ =Hˆ 2−Hˆ2, we need the expectation Setting f (0) := 0, the function f is everywhere defined ˆ 2 ˆ 2 = 4 ∂4 = ˆ 2= R 4 value of H .FromH 4m2 ∂x4 0, we have H on and “smooth”. Further—unlike plane waves, a ∗(x)Hˆ 2(x)dx = 0. Since Hˆ2 is strictly positive, x=−a delta functions and the like—it is square-integrable 2 +∞ − − 1 the variance Hˆ is negative, which obviously is impossi- = 3 2x2 = and normalised: x∈R f (x) dx x=0 x e dx − 1 +∞ − 1 ble. What is wrong here? 2x2 = = 4x2 e x=0 1. For the derivative of g(x): e we find = − 1 −2 · = 1 −3 > g (x) ( 4 x ) g(x) 2 x g(x), thus for x 0, there 2.1.3 Eigenvalues of hermitian operators I holds −3 − / 3 − / − / x Consider a radially-symmetric potential V (r )inthree xˆ3pxˆ 3 2g(x) = x3 − x 5 2 + x 3 2 g(x) and i 2 2 space-dimensions. Using the ansatz (r ) = R(r)Y (ϑ, ϕ) lm (3) in spherical coordinates we get the spherical harmon- u(r) ics Y and by substituting R(r) = we get a radial − / d / lm r pˆxˆ3x 3 2g(x) = x3 2g(x) Schrodinger¨ equation for u(r) by adding a centrifugal i dx 2 + potential l(l 1) to V (r). We thus have a radial position −3 2mr 3 1/2 3/2 x = ∂ = x + x g(x). (4) operator rˆ and a corresponding momentum pˆr i ∂r , i 2 2 which fulfil the canonical commutator relation. Now, pˆr i α is hermitian, but ψ(r) = e r is a normalisable eigen- As f is even, and each power of xˆ and pˆ changes sym- ˆ = metries once, there holds Af i f for the function as a whole. Altogether, Aˆ is a hermitian operator with an 2 For these and other related examples see the articles by Gieres eigenfunction f ,butitseigenvalueisnotarealnumber. [4] and Bonneau et al. [5], which are very much recommended Where is the error? to the reader. 3 For a real commutator (without the imaginary prefactor i), 4 It can be arbitrarily often differentiated. It should not bother us one can use the ladder operators in a harmonic oscillator with that as a complex function there is an essential singularity in [aˆ, aˆ†] = 1 in a similar fashion. the origin, since we only consider real functions here. C 2015 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim Wiley Online Library 645 Fortschritte der Physik K. S. Ranade: Functional analysis and quantum mechanics—an introduction Progress of Physics sum is, i. e. in the course of calculation we secretly leave 0.7 the Hilbert space for some time, which is not allowed. 0.6 Review Paper 0.5 2.3 Hilbert spaces 0.4 We start with the definition of a Hilbert space. We will use 0.3 Dirac’s bra-ket notation when we find it appropriate. 0.2 Definition 1 (Hilbert space). A pre-Hilbert space is a vec- tor space over some field K equipped with a scalar product 0.1 (or inner product), i. e. a function ·|·: V × V → Kwith the following three properties: 2 4 1. sesquilinearity: x|λv + μw=λx|v+μx|w and Figure 1 A wavefunction with imaginary eigenvalue. λv + μw|x=λ∗v|x+μ∗w|x, 2. anti-symmetry: w|v=v|w∗ and 2.2 Linear algebra and functional analysis 3.