Selfadjoint extensions in quantum mechanics springerlink. We will try to o er a brief account of some recent ideas in the theory of self adjoint extensions of symmetric operators on hilbert spaces and their applications to a few speci c problems in quantum mechanics. Self adjoint extensions of operators and the teaching of quantum mechanics. This manuscript provides a brief introduction to mathematical methods in quantum mechanics spectral theory with applications to schrodinger operators. On selfadjoint extensions and symmetries in quantum mechanics.
Selfadjoint extensions and susy breaking in supersymmetric. The necessary mathematical background is then built by developing the theory of selfadjoint extensions. Use features like bookmarks, note taking and highlighting while reading selfadjoint extensions in quantum mechanics. Additional physical requirements such as parity, time reversal, and positivity are used to restrict the large class of self adjoint extensions of the hamiltonian.
May 14, 2007 consequently, the standard technology of regular quantum mechanics is supplemented by a mandatory regularization toolbox. Applications of selfadjoint extensions in quantum physics. On z invariant selfadjoint extensions of the laplacian. Quantum physics by michel le bellac cambridge core. Jun 08, 2007 constructing quantum observables and self adjoint extensions of symmetric operators. A unique discussion of mathematical methods with applications to quantum mechanics nonselfadjoint operators in quantum physics. Citeseerx document details isaac councill, lee giles, pradeep teregowda.
The first part covers mathematical foundations of quantum mechanics from selfadjointness, the spectral theorem, quantum dynamics including stones and the rage theorem to perturbation. Self adjoint extensions of operators and the teaching of quantum mechanics guy bonneau jacques faraut y galliano valent abstract for the example of the in nitely deep well potential, we point out some paradoxes which are solved by a careful analysis of what is a truly self adjoint operator. We show that only for two particular sae, whose domains are scale invariant, the algebra of n2 susy is realized, one with manifest susy and the other with. However, many physical problems are formulated as a time. An analysis of the invariance properties of self adjoint extensions of symmetric operators under the action of a group of symmetries is presented. The self adjoint operator is actually the closure of h.
Citeseerx selfadjoint extensions and susy breaking in. Selfadjoint matrix an overview sciencedirect topics. Self adjoint extensions in quantum mechanics begins by considering quantization problems in general, emphasizing the nontriviality of consistent operator construction by presenting paradoxes of the nave treatment. Pseudohermitian quantum mechanics with unbounded metric. Pdf on selfadjoint extensions and symmetries in quantum. In most french universities, quantum mechanics is usually taught in the third year courses. Selfadjoint extensions of operators in quantum mechanics.
Voronov 20120427 science general theory and applications to schrodinger and dirac equations with singular potentials. Selfadjoint extensions of operators and the teaching of quantum. Furthermore, we discuss the cartesian and polar decomposition of arbitrary normal operators and the possibility to operationalize the corresponding observables. Linear operators for quantum mechanics available for download and read online in other formats.
It is very common to find singular hamiltonians in quantum mechanics, especially in quantum systems in the presence of topological defects, which are usually modeled by point interactions. Self adjoint operators download ebook pdf, epub, tuebl, mobi. Quadratic forms and the friedrichs extension 67 x2. Constructing quantum observables and selfadjoint extensions. Selfadjoint extensions in quantum mechanics in quantum mechanics, observables correspond to selfadjoint operators. Selfadjoint extensions in quantum mechanics general theory.
Given a unitary representation of a lie group g on a hilbert space h, we develop the theory of ginvariant selfadjoint extensions of symmetric. The first one is that nonstandard singular selfadjoint extensions x 2, x 4 can be described by the positiondependent mass hamiltonian effectivemass hamiltonian. On selfadjoint extensions and symmetries in quantum mechanics 3 not self adjoint. Of particular importance is the existence, and sometimes explicit constructions, of self adjoint extensions. Here, as in any physical theory, we have to consider only those quantities which may be measured at least in principle. Selfadjoint extensions to the dirac coulomb hamiltonian. L download it once and read it on your kindle device, pc, phones or tablets. The energy eigenvalues obtained by solving this equation must be real one of the axioms of quantum mechanics is that physical observables, in this case energy, correspond to self adjoint operators, in this case the hamiltonian operator hi, acting on the hilbert space 7h which describes the system in question. On zinvariant selfadjoint extensions of the laplacian on. The algebra of selfadjoint operators in spectral theory with. Frontiers selfadjoint extension approach for singular. Going through the quantum mechanics book by capri, am time and again held with some stupid doubts on this topic of self adjointness.
Some consequences are worked out, which could lead to experimental checks. On selfadjoint extensions and symmetries in quantum mechanics on selfadjoint extensions and symmetries in quantum mechanics ibort, alberto. Physics stack exchange is a question and answer site for active researchers, academics and students of physics. Mar 28, 2001 for the example of the infinitely deep well potential, we point out some paradoxes which are solved by a careful analysis of what is a truly self adjoint operator. In functional analysis, one is interested in extensions of symmetric operators acting on a hilbert space. General theory and applications to schrodinger and dirac equations with singular potentials progress in mathematical physics 2012 by d. This problem arises, for example, when one needs to specify domains of self adjointness for formal expressions of observables in quantum mechanics. Only in this case a unitary evolution of the system is given. General theory and applications to schrodinger and dirac equations with singular potentials. A unified smatrix framework of quantum singular interactions is presented for the comparison of self adjoint extensions and physical renormalization.
We illustrate situations for which different boundary conditions lead to different operators and hence to different physics. Selfadjoint extensions in quantum mechanics general. Everyday low prices and free delivery on eligible orders. This allows the construction of exactly solvable models in quantum mechanics, quantum field theory, high energy physics, solidstate physics, microelectronics and other fields. Selfadjoint extensions in quantum mechanics ebook by d.
Selfadjoint extensions in quantum mechanics begins by considering quantization problems in general, emphasizing the nontriviality of consistent operator construction by presenting paradoxes of the naive treatment. The first part covers mathematical foundations of quantum mechanics from self adjointness, the spectral theorem, quantum dynamics including stones and the rage theorem to perturbation theory for self adjoint operators. To construct a selfadjoint operator the domain of the operator has to be specified by imposing an appropriate boundary condition or conditions on the wave functions on which the operator acts. This allows the construction of exactly solvable models in quantum mechanics, quantum field theory, high energy. It then builds the necessary mathematical background following it by the theory of self adjoint extensions. This exposition is devoted to a consistent treatment of quantization problems, based on appealing to. This new text comes closest to the contemporary heart of the discipline and i cannot recommend it strongly enough to everyone interested in quantum mechanics, and especially students and teachers this text is the best one available for a course designed to expose students to the role played by quantum mechanics in todays world. By stones theorem on oneparameter unitary groups, self adjoint operators are precisely the infinitesimal generators of unitary groups of time evolution operators. Our goal in this note is to prove the following sharp version of ehrenfests theorem for general selfadjoint operators.
For a given group g, criteria for the existence of ginvariant selfadjoint extensions of the laplacebeltrami operator over a riemannian manifold are illustrated and critically revisited. We also analyze the relation between the reduction theory of the unitary representation and the reduction of the ginvariant unbounded operator. We give a self contained presentation of the theory of self adjoint extensions using the technique of boundary triples. Otherwise, we go immediately to a kind of scholastic.
These atoms can be described by a coulomb potential, vr ze 2 r, where z is the number of protons in the nucleus and e is the elementary charge. On the inequivalence of renormalization and selfadjoint. X 2interaction corresponds to the massjump profile see fig. Borchers, jakob yngvason, local nets and self adjoint extensions of quantum field operators, letters in mathematical physics web. A description of the spectra of self adjoint extensions in terms of the corresponding krein maps weyl functions is given. An analysis of the invariance properties of selfadjoint extensions of symmetric operators under the action of a group of symmetries is presented. Both methods are based on the selfadjoint extension approach. The reality of the energy eigenvalues becomes important when examining hydrogenic atoms using the dirac equation. Oct 26, 2014 on selfadjoint extensions and symmetries in quantum mechanics on selfadjoint extensions and symmetries in quantum mechanics ibort, alberto. The shared purpose in this collection of papers is to apply the theory of self adjoint extensions of symmetry operators in various areas of physics. This exposition is devoted to a consistent treatment of quantization problems, based on appealing to some nontri. Citeseerx on possible extensions of quantum mechanics.
Why are the operators in quantum mechanics in general unbounded. Guy bonneau, jacques faraut, galliano valent, self adjoint extensions of operators and the teaching of quantum mechanics, arxiv. Guy bonneau, jacques faraut, galliano valent submitted on 28 mar 2001. Presentation outline the role of the self adjoint operators in quantum mechanics self adjoint extension of an operator classical precursors of self adjoint extension simple example. We also apply the apparatus of self adjoint extensions and renormalization to clarify aspects of anomalous symmetry breaking, supersymmetric quantum mechanics, and onedimensional quantum mechanics. Pdf linear operators for quantum mechanics download full. In mathematics, a selfadjoint operator or hermitian operator on a finitedimensional complex vector space v with inner product. The description of such systems is not complete until a self adjoint extension of the operator has been determined, e.
Download it once and read it on your kindle device, pc, phones or tablets. Mathematical aspects presents various mathematical constructions influenced by quantum mechanics and emphasizes the spectral theory of nonadjoint operators. Selfadjoint extensions of operators and the teaching of. Again, the physical condition that the expectation values of observables must be real numbers demands that we identify the observables of this system with the self adjoint operators acting in. We then describe the self adjoint extensions and their spectra for the momentum and the hamiltonian operators in different settings. It can easily be shown that self adjoint operators must have real eigenvalues. Dhx be densely defined selfadjoint operators on a hilbert space x. A sharp version of ehrenfests theorem for general self. There are however densely defined symmetric operators that admit more than one self adjoint extension, or none at all. Therefore, is a self adjoint extension of h, and the pair defines a unitary quantum system. D b, and in the domain of b the operators are equals, then we say that a is an extension of b. Download pdf linear operators for quantum mechanics book full free. We then describe the self adjoint extensions and their spectra for the momentum and the hamiltonian operators in different physical situations.
We propose to generalize the probability axiom of quantum mechanics to self adjoint positive operators of trace one. General theory and applications to schrodinger and dirac equations with singular potentials progress in mathematical physics book 62. On selfadjoint extensions and symmetries in quantum. In quantum mechanics the observables are supposed to be selfadjoint operators, in particular the hamiltonian. Applications include quantum graphs, point interactions, hybrid spaces and singular perturbations. Self adjoint extensions and susy breaking in supersymmetric quantum mechanics by h. Selfadjoint extensions of operators and the teaching of quantum mechanics guy bonneau jacques faraut y galliano valent abstract for the example of the in nitely deep well potential, we point out some paradoxes which are solved by a careful analysis of what is a truly self adjoint operator. By stones theorem on oneparameter unitary groups, selfadjoint operators are precisely the infinitesimal generators of unitary groups of time evolution operators. We show that only for two particular sae, whose domains are scale invariant, the algebra of n 2 susy is realized, one with manifest susy. Self adjoint extensions in quantum mechanics in quantum mechanics, observables correspond to self adjoint operators. Read selfadjoint extensions in quantum mechanics general theory and applications to schrodinger and dirac equations with singular potentials by d.
Selfadjoint extensions of operators and the teaching of quantum mechanics. Request pdf selfadjoint extensions in quantum mechanics. For the longrange conformal interaction the two methods are not equivalent, with renormalization acting as selector of a preferred extension and regulator of the unbounded hamiltonian. The necessary mathematical background is then built by developing the theory of self adjoint extensions. Self adjoint extensions in quantum mechanics begins by considering quantization problems in general, emphasizing the nontriviality of consistent operator construction by presenting paradoxes of the naive treatment. If v is finitedimensional with a given orthonormal basis, this is equivalent to the condition that the matrix of a is a hermitian matrix, i. Massjump and massbump boundary conditions for singular self. Basics of the theory of selfadjoint extensions of symmetric operators. Selfadjoint extensions in quantum mechanics springer. By considering several problems such as the onedimensional calogero. General theory and applications to schrodinger and dirac equations with singular potentials progress in mathematical physics book 62 ebook.
Beltrami operator over a riemannian manifold are illustrated and critically revisited. To find the adjoint action of linear combinations of operators is formally easy if you know the adjoint of the components, as it is seen from the definition. We consider the self adjoint extensions sae of the symmetric supercharges and hamiltonian for a model of susy quantum mechanics in with a singular superpotential. Guy bonneau, jacques faraut, galliano valent download. Use features like bookmarks, note taking and highlighting while reading self.
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