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7. Quantization of the Harmonic Oscillator – Ariadne's Thread In 7. Quantization of the Harmonic Oscillator – Ariadne’s Thread in Quantization Whoever understands the quantization of the harmonic oscillator can un- derstand everything in quantum physics. Folklore Almost all of physics now relies upon quantum physics. This theory was discovered around the beginning of this century. Since then, it has known a progress with no analogue in the history of science, finally reaching a status of universal applicability. The radical novelty of quantum mechanics almost immediately brought a conflict with the previously admitted corpus of classical physics, and this went as far as rejecting the age-old representation of physical reality by visual intuition and common sense. The abstract formalism of the theory had almost no direct counterpart in the ordinary features around us, as, for instance, nobody will ever see a wave function when looking at a car or a chair. An ever-present randomness also came to contradict classical determinism.1 Roland Omn`es, 1994 Quantum mechanics deserves the interest of mathematicians not only be- cause it is a very important physical theory, which governs all microphysics, that is, the physical phenomena at the microscopic scale of 10−10m, but also because it turned out to be at the root of important developments of modern mathematics.2 Franco Strocchi, 2005 In this chapter, we will study the following quantization methods: • Heisenberg quantization (matrix mechanics; creation and annihilation operators), • Schr¨odinger quantization (wave mechanics; the Schr¨odinger partial differential equation), • Feynman quantization (integral representation of the wave function by means of the propagator kernel, the formal Feynman path integral, the rigorous infinite- dimensional Gaussian integral, and the rigorous Wiener path integral), • Weyl quantization (deformation of Poisson structures), 1 From the Preface to R. Omn`es, The Interpretation of Quantum Mechanics, Princeton University Press, Princeton, New Jersey, 1994. Reprinted by permis- sion of Princeton University Press. We recommend this monograph as an intro- duction to the philosophical interpretation of quantum mechanics. 2 F. Strocchi, An Introduction to the Mathematical Structure of Quantum Me- chanics: A Short Course for Mathematicians, Lecture Notes, Scuola Normale, Pisa (Italy). Reprinted by permission of World Scientific Publishing Co. Pte. Ltd. Singapore, 2005. 428 7. Quantization of the Harmonic Oscillator • Weyl quantization functor from symplectic linear spaces to C∗-algebras, • Bargmann quantization (holomorphic quantization), • supersymmetric quantization (fermions and bosons). We will choose the presentation of the material in such a way that the reader is well prepared for the generalizations to quantum field theory to be considered later on. Formally self-adjoint operators. The operator A : D(A) → X on the complex Hilbert space X is called formally self-adjoint iff the operator is linear, the domain of definition D(A) is a linear dense subspace of the Hilbert space X, and we have the symmetry condition χ|Aϕ = Aχ|ϕ for all χ, ψ ∈ D(A). Formally self-adjoint operators are also called symmetric operators. The following two observations are crucial for quantum mechanics: • If the complex number λ is an eigenvalue of A, that is, there exists a nonzero element ϕ ∈ D(A) such that Aϕ = λϕ,thenλ is a real number. This follows from λ = ϕ|Aϕ = Aϕ|ϕ = λ†. • If λ1 and λ2 are two different eigenvalues of the operator A with eigenvectors ϕ1 and ϕ2,thenϕ1 is orthogonal to ϕ2. This follows from (λ1 − λ2)ϕ1|ϕ2 = Aϕ1|ϕ2−ϕ1|Aϕ2 =0. In quantum mechanics, formally self-adjoint operators represent formal observables. For a deeper mathematical analysis, we need self-adjoint operators, which are called observables in quantum mechanics. Each self-adjoint operator is formally self-adjoint. But, the converse is not true. For the convenience of the reader, on page 683 we summarize basic material from func- tional analysis which will be frequently encountered in this chapter. This concerns the following notions: formally adjoint operator, adjoint operator, self-adjoint oper- ator, essentially self-adjoint operator, closed operator, and the closure of a formally self-adjoint operator. The reader, who is not familiar with this material, should have a look at page 683. Observe that, as a rule, in the physics literature one does not distinguish between formally self-adjoint operators and self-adjoint operators. Peter Lax writes:3 The theory of self-adjoint operators was created by John von Neumann to fashion a framework for quantum mechanics. The operators in Schr¨odin- ger’s theory from 1926 that are associated with atoms and molecules are partial differential operators whose coefficients are singular at certain points; these singularities correspond to the unbounded growth of the force between two electrons that approach each other. I recall in the summer of 1951 the excitement and elation of von Neumann when he learned that Kato (born 1917) has proved the self-adjointness of the Schr¨odinger oper- ator associated with the helium atom.4 3 P. Lax, Functional Analysis, Wiley, New York, 2003 (reprinted with permis- sion). This is the best modern textbook on functional analysis, written by a master of this field who works at the Courant Institute in New York City. For his fundamental contributions to the theory of partial differential equations in mathematical physics (e.g., scattering theory, solitons, and shock waves), Peter Lax (born 1926) was awarded the Abel prize in 2005. 4 J. von Neumann, General spectral theory of Hermitean operators, Math. Ann. 102 (1929), 49–131 (in German). 429 And what do the physicists think of these matters? In the 1960s Friedrichs5 met Heisenberg and used the occasion to express to him the deep gratitude of the community of mathematicians for having created quantum mechan- ics, which gave birth to the beautiful theory of operators in Hilbert space. Heisenberg allowed that this was so; Friedrichs then added that the math- ematicians have, in some measure, returned the favor. Heisenberg looked noncommittal, so Friedrichs pointed out that it was a mathematician, von Neumann, who clarified the difference between a self-adjoint operator and one that is merely symmetric.“What’s the difference,” said Heisenberg. As a rule of thumb, a formally self-adjoint (also called symmetric) differential op- erator can be extended to a self-adjoint operator if we add appropriate boundary conditions. The situation is not dramatic for physicists, since physics dictates the ‘right’ boundary conditions in regular situations. However, one has to be careful. In Problem 7.19, we will consider a formally self-adjoint differential operator which cannot be extended to a self-adjoint operator. The point is that self-adjoint operators possess a spectral family which al- lows us to construct both the probability measure for physical observables and the functions of observables (e.g., the propagator for the quantum dy- namics). In general terms, this is not possible for merely formally self-adjoint operators. The following proposition displays the difference between formally self-adjoint and self-adjoint operators. Proposition 7.1 The linear, densely defined operator A : D(A) → X on the com- plex Hilbert space X is self-adjoint iff it is formally self-adjoint and it always follows from ψ|Aϕ = χ|ϕ for fixed ψ, χ ∈ X and all ϕ ∈ D(A) that ψ ∈ D(A). Therefore, the domain of definition D(A) of the operator A plays a critical role. The proof will be given in Problem 7.7. Unitary operators. As we will see later on, for the quantum dynamics, unitary operators play the decisive role. Recall that the operator U : X → X is called unitary iff it is linear, bijective, and it preserves the inner product, that is, Uχ|Uϕ = χ|ϕ for all χ, ϕ ∈ X. This implies ||Uϕ|| = ||ϕ|| for all ϕ ∈ X. Hence ||U|| := sup ||Uϕ|| =1 ||ϕ||≤1 if we exclude the trivial case X = {0}. The shortcoming of the language of matrices noticed by von Neu- mann. Let A : D(A) → X and B : D(B) → X be linear, densely defined, formally J. von Neumann, Mathematical Foundations of Quantum Mechanics (in Ger- man), Springer, Berlin, 1932. English edition: Princeton University Press, 1955. T. Kato, Fundamental properties of the Hamiltonian operators of Schr¨odinger type, Trans. Amer. Math. Soc. 70 (1951), 195–211. 5 Schr¨odinger (1887–1961), Heisenberg (1901–1976), Friedrichs (1902–1982), von Neumann (1903–1957), Kato (born 1917). 430 7. Quantization of the Harmonic Oscillator self-adjoint operators on the infinite-dimensional Hilbert space X.Letϕ0,ϕ1,ϕ2,... be a complete orthonormal system in X with ϕk ∈ D(A) for all k.Set ajk := ϕj |Aϕk j, k =0, 1, 2,... The way, we assign to the operator A the infinite matrix (ajk). Similarly, for the operator B, we define bjk := ϕj |Bϕk j, k =0, 1, 2,... Suppose that the operator B is a proper extension of the operator A.Then ajk = bjk for all j, k =0, 1, 2,..., but A = B. Thus, the matrix (ajk) does not completely reflect the properties of the operator A. In particular, the matrix (ajk) does not see the crucial domain of definition D(A) of the operator A. Jean Dieudonn´ewrites:6 Von Neumann took pains, in a special paper, to investigate how Hermitean (i.e., formally self-adjoint) operators might be represented by infinite ma- trices (to which many mathematicians and even more physicists were sen- timentally attached) . Von Neumann showed in great detail how the lack of “one-to-oneness” in the correspondence of matrices and operators led to their weirdest pathology, convincing once for all the analysts that matrices were a totally inadequate tool in spectral theory. 7.1 Complete Orthonormal Systems A complete orthonormal system of eigenstates of an observable (e.g., the energy operator) cannot be extended to a larger orthonormal system of eigenstates.
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