The Time-Dependent Schrodinger Equation: The …

1 178 Brazilian Journal of Physics, vol. 38, no. 1, March, 2008 The Time-Dependent Schr odinger Equation: The Need for the Hamiltonian to be Self-AdjointVanilse S. Araujo,Escola de Engenharia Maua, S ao Paulo, Brazil and Faculdade de Engenharia da Fundac ao Santo Andre, S ao Paulo, BrazilF. A. B. Coutinho,Faculdade de Medicina da Universidade de S ao Paulo, S ao Paulo, 01246-903, BrazilF. M. ToyamaDepartment of Information and Communication Sciences, Kyoto Sangyo University, Kyoto 603-85555 JapanReceived on 20 December, 2007We present some simple arguments to show that quantum mechanics operators are required to be emphasize that the very definition of a self-adjoint operator includes the prescription of a certain domain ofthe operator. We then use these concepts to revisit the solutions of the Time-Dependent Schroedinger equation ofsome well-known simple problems the infinite square well, the finite square well, and the harmonic show that these elementary illustrations can be enriched by using more general boundary conditions, whichare still compatible with self-adjointness.

2 In particular, we show that a puzzling problem associated with theHydrogen atom in one dimension can be clarified by applying the correct requirements of then come to Stone s theorem, which is the main topic of this paper, and which is shown to relate theusual definitions of a self-adjoint operator to the possibility of constructing well-defined solutions of the Time-Dependent Schr odinger : Operator domains; Self-adjointness; Stone theorem; Quantum Mechanics; Operator exponentialI. INTRODUCTIONIn quantum mechanics observables are represented by op-erators acting on the functions belonging to the Hilbert spaceof the system under consideration [1]. However, in contrastto the mathematical literature, where operators are defined bytheir action (that is, what they do to the functions on whichthey operate) and by their domain (that is, the set of functionson which they operate), in the physical literature domains arealmost never mentioned and operators are defined only bytheir actions.

3 Operators in infinite dimensional Hilbert spaceare not defined for all the functions of the space, and this sug-gests that one should be aware of situations where domainsare reason why the domains of the operators are so impor-tant, even in physics, is that we need the operators in quan-tum mechanics to be self-adjoint and operators are self-adjointonly in well-defined and prescribed domains. But why is itso important for operators in quantum mechanics to be self-adjoint? There are two important reasons: The first one is thatthe eigenvalues are real and the eigenfunctions form a com-plete set of orthogonal functions so that any function of theHilbert space of the system in consideration can be expandedin this set. (The reason for the quotation marks in the wordfunctions will become clear later.) The second reason is thatonly if the HamiltonianHof the system is a self-adjoint op-erator, the Time-Dependent Schr odinger equationi t =H has a unique solution such that (t) 2= (0) 2 for alltimes.

4 This is the content of Stone s theorem [2] that we willexplain in detail in section 3, and has to do with the possibilityof constructing the exponential of an operator [3], in this casethe operator exp[ i Ht]. However, it is hard to see how thosetwo properties are linked with domains. The objective of thispaper is to make this link intuitively clearUntil quite recently, domains or self-adjointness were notmentioned in the physical literature. However, in the last fewyears or so, some articles [4], [5], [6], [7] in the physics peda-gogical literature begun to point out examples where domainsof operators are essential to the full solution of the problemsposed. We are aware of only three pedagogical articles pub-lished before those articles [6-9] that mention domains andself-adjointness. One is by Jordan [8], another by Capri [9],and finally an article by Zhu and Klauder [10] that relates lackof self -adjointness with strange classical is interesting to try to understand why it is possible toneglect domains in the physical literature.

5 First, it is truethat domains rarely cause problems. This is indeed so, butGieres [11] cites seven examples of physical puzzles causedby the manipulation of operators neglecting attention to theirdomains. Second, if domains are not mentioned, it is naturalto think that domains are automatically specified. One tendsto think that if Ais an operator and A belongs to the Hilbertspace of the system under consideration then belongs to thedomain of the operator. It is true for that for to belong to thedomain of an operator A, A must belong to the Hilbert spaceof the system and we will assume this in the remaining of thepaper without further comment. However, although there areoperators with good properties in such a large domain (seeAppendix 1 and the example ( ) and ( ) below) in generalit is necessary to restrict the domain by specifying boundaryconditions to be obeyed by the functions of the domain. Thedomains are specified in such a way, but it is hard to see thatthis has been done because this fact is never mentioned.

6 As wewill see domains are specified by the boundary conditions im-posed when solving the time -independent Schr odinger S. Araujo, F. A. B. Coutinho, and F. M. Toyama179 This paper is organized as follows. In the next section werecapitulate briefly, and in a non- technical way, what a self-adjoint operator is, and apply self-adjointness to the solutionof the time -independent Schr odinger equation. The exam-ples chosen to illustrate the concept of self- adjointness areall the familiar examples found in quantum mechanics text-books: the infinite square well, the harmonic oscillator, thehydrogen atom, etc. However, it is shown how the concept ofself-adjoint extension enriches these familiar illustrations, andclarifies also a puzzling situation (the hydrogen atom in onedimension). Next, in section 3, we state Stone s theorem andin section 4 we show how this theorem is related to the possi-bility of constructing the exponential of an operator emphasiz-ing how this construction is linked to the domain of the opera-tor.

7 This completes our main task in this paper that is to showwhy self-adjointness is essential to operators in quantum me-chanics. In the Appendix, for completeness, we present twoexamples the harmonic oscillator and the finite square well of Hamiltonians that do not require boundary conditions forthe definition of their domains. Finally, in Appendix 2, wededuce an unusual boundary condition atr=0 for the s-waveradial part of the Hamiltonian describing the hydrogen SELF-ADJOINTNESS AND THE time -INDEPENDENTSCHR ODINGER EQUATIONIn this paper we will confine ourselves, for definiteness, toone-dimensional systems. Assume a particle that is free tomove in the entire real line. Then, the Hilbert space of thissystem is the set of functions such that + (x) (x)dx= + | (x)|2dx= 2=finite,(1)where (x)denotes the complex conjugate of (x). Theinner product between two function 1(x)and 2(x)in thisspace is defined by< 1| 2>= + 1(x) 2(x)dx.(2)Given a certain operator Owe define its Hermitian conjugateas the operator O defined by< 1|( O 2)>= + 1(x)( O 2(x))dx= + ( O 1(x)) 2(x)dx=<( O 1)| 2>.

8 (3)When the operators O and Ohave the same action (that isthey do the same thing to the functions on which they operate)and equation (3) holds, they are called Hermitian (symmetricby mathematicians). If in addition they have the same domainthen they are self-adjoint. The distinction between Hermitianoperators and self-adjoint operators is rather subtle. To clarifythe distinction reference [5] presents two examples of opera-tors with the same action but with different domains so that inthe first example the operator is Hermitian and in the secondthe operator is self-adjoint. (One should note that the distinc-tion between Hermitian and self-adjoint operators disappearsfor bounded operators which are defined for all the functionsof the Hilbert space and hence for operators in finite dimen-sional spaces, that is for matrices.)As mentioned in the introduction, when solving the time -independent Schr o-dinger equation, it is in general necessaryto restrict the domains of operators by imposing boundaryconditions so that they become self-adjoint.

9 In the remain-ing of this section we present examples of this procedure. Theexamples ( ) and ( ) are illustrations of operators wherethe domains are restricted only by their action. The other ex-amples are illustrations of cases where boundary conditionsare necessary in addition to the restrictions imposed by )The operator xIn this example we consider the operator x,that is multi-plication byx, acting on the Hilbert space of the functionsgiven by equation (1). A natural domain for this operator isthe set of functions (x)that in addition to having finite normalso obeys + |x (x)|2dx=finite.(4)Is this operator so defined self-adjoint? Consider two func-tions 1(x)and 2(x)obeying equation (4). We have that + 1(x)(x 2(x))dx= + (x 1(x)) 2(x)dx,(5)and this happens if both 1(x)and 2(x)obey equation (4).So the operator xis self- adjoint in this natural )A criterion of self-adjointnessIt is generally difficult to check domains. So we presentbelow a result that helps to recognize if an operator is self-adjoint in its domain.

10 This result is explained in reference180 Brazilian Journal of Physics, vol. 38, no. 1, March, 2008[5], and is a basic criterion of self-adjointness. To see if anoperator Ois self-adjoint consider the equations O (x)= i (x),(6)where is a constant which maintains the dimensional consis-tence of the equations . If the above equations have no squareintegrable solutions in the proposed domain then the operatoris self-adjoint. Clearly the equations x (x)=x (x)= ix0 (x)(7)have no solution, except (x) 0, in the entire Hilbert space.(The Dirac delta function is not admissible because it is notsquare integrable [12].) So the operator xis self-adjoint inits natural domain as shown in the previous example )The operator momentumThe operator momentum, p= iddx, acting on the space offunctions given by equations (1) is self-adjoint in its natural domain, that is, the domain are functions (x)such that p (x)are square integrable (satisfies equation (1)). This can be eas-ily deduced from the fact that equation (6) with Oreplaced by phas no square integrable solutions in this next two examples (examples and ) showhow the specification of the domains by imposing boundaryconditions enriches even the most common examples found inquantum mechanics textbooks.