Such an unexpected connection is outlined in the final part of the paper. Rigorously speaking, to identify the internuclear distance by the x variable of equation (13) would imply the inclusion of an unphysical region corresponding to negative values of the internuclear distance. a complete set, even if we don't happen to know them. Two of these potentials are one-dimensional (1D henceforth), precisely the Morse and the Pöschl-Teller potentials. The starting point is, needless to say, the energy functional that, on taking equation (42) into account, can be recast as follows: where the symbol t stands for \tanh \alpha x,\alpha being defined again by equation (15). In all introductory quantum mechanics textbooks, it is customarily presented as an invaluable technique aimed at finding approximate estimates of ground state energies [3–7]. In a monumental review paper published at the very beginning of the fifties [17], Infeld and Hull presented a systematic study about all possible 1D potentials for which the corresponding stationary Schrödinger equation can be exactly factorized. This should help students to appreciate how some basic features of a phenomenon can sometimes be grasped even by using idealized, nonrealistic models. Compared to perturbation theory, the variational method can be more robust in situations where it's hard to determine a good unperturbed Hamiltonian (i.e., one which makes the perturbation small but is still solvable). with χ, of course, being the solution of equation (37). Before continuing, the teacher should advise his/her students that the quantity in the rhs of equation (5) is a mathematical object called functional and that the branch on mathematics that studies the properties of functionals, the calculus of variations, is a rather advanced topic. good unperturbed Hamiltonian, perturbation theory can be more was proposed in 1929 by Morse [8] as a simple analytical model for describing the vibrational motion of diatomic molecules. The first integral into the rhs of equation (17) is expanded to have. The chapter describes the variational method and gives a simple example of how it is used to estimate eigenenergies and eigenfunctions. A 'toy' model for the Morse potential. The two approximation methods described in this chapter‐the variational method and the perturbation method‐are widely used in quantum mechanics, and has applications to other disciplines as well. Moreover, on further letting x\to \alpha x, after simple algebra equation (14) can be recast as follows: Figure 1. Here we review three approximate methods each of On the other hand, in cases where there is a The knowledge of higher-order eigenstates would require mathematical techniques that are out of the limits and the scopes of the present paper. if the following condition: It could be worth proposing to students an intuitive interpretation of the inequality (24), which I took from an exercise in the Berkeley textbook [1]. We are not aware of previous attempts aimed at providing a variational route to factorization. The Variational Method James Salveo L. Olarve PHYDSPHY, DLSU-M January 29, 2010 2. Factorization was introduced at the dawn of quantum mechanics by Schrödinger and by Dirac as a powerful algebraic method to obtain the complete energy spectrum of several 1D quantum systems. Volume 39, If you have a user account, you will need to reset your password the next time you login. View the article online for updates and enhancements. as a trial function for the wavefunction for the problem, which consists of some adjustable By continuing to use this site you agree to our use of cookies. Variational Method. It is a useful analytical model to describe finite potential wells as well as anharmonic oscillators, and is sketched in figure 4. Related terms: Configuration Interaction; Hamiltonian; Perturbation Theory [Alpha] Wave Function; Symmetry No. Consider the 1D motion of a mass point m under the action of a conservative force which is described via the potential energy function U(x). This allows calculating approximate wavefunctions such as molecular orbitals. 2 To Franco Gori, on his eightieth birthday. where ∇2() denotes the Laplacian operator acting on the stationary states u=u({\boldsymbol{r}}), with {\boldsymbol{r}} denoting the electron position vector with respect to the nucleus. From equation (10) it follows that the oscillator energy cannot assume values less than ω/2 (when expressed through physical units). But there is more. The variational method is an approximate method used in quantum mechanics. Subsequently, three celebrated examples of potentials will be examined from the same variational point of view in order to show how their ground states can be characterized in a way accessible to any undergraduate. We know the ground state energy of the hydrogen atom is -1 Ryd, or -13.6 ev. The variational method was the key ingredient for achieving such a result. From equation (55), on again taking equation (52) into account, it follows that the energy of the ground state is just 1. Equation (5) will be the starting point of our analysis. Accordingly, on using the transformations kx → x and E/U0 → E, it is immediately proved that the energy functional (5) becomes, the dimensionless parameter α being defined by. To this end, we will illustrate a short 'catalogue' of several celebrated potential distributions for which the ground state can be found without actually solving the corresponding Schrödinger equation, but rather through a direct minimization of an energy functional. The final example we wish to offer is a simple and compact determination of the ground state of the hydrogen atom. lengths and energies will again be measured in terms of U0 and α/k, respectively. efficient than the variational method. To this end, consider the following differential operator: where the first 'factor' has been obtained simply by changing the sign of the derivative operator {\rm{d}}/{\rm{d}}x within the factor of equation (52). In figure 1 a graphical representation of the Morse potential is sketched. This is because there exist highly entangled many-body states that Actually the potential in equation (30) is customarily named hyperbolic Pöschl-Teller potential, and was first considered by Eckart as a simple continuous model to study the penetration features of some potential barriers [9]. Lett. Therefore, we need to opt for appropriate approximations when facing the remaining vast majority of quantum-mechanical problems. is the one with the lowest energy? A382, 1472 (2018)]. This makes our approach particularly suitable for undergraduates. adjusted until the energy of the trial wavefunction is minimized. function J. Phys. There exist only a handful of problems in quantum mechanics which can be solved exactly. This method is free of such essential diffi- culty as the necessity of knowing the entire spectrum of the unperturbed problem, and makes it possible to make estimates of the accuracy of variational calcula- tions. Functional minimization requires the knowledge of mathematical techniques that cannot be part of undergraduate backgrounds. where it will be tacitly assumed henceforth that any integration has to be carried out across the whole real axis (-\infty ,+\infty ). A pictorial representation of the Rosen-Morse potential in equation (42). In quantum mechanics, the variational method is one way of finding approximations to the lowest energy eigenstate or ground state, and some excited states. Note that, in order for the function in equation (23) to represent a valid state, it is necessary that the arguments of both exponentials be negative, which occurs only if α < 2, i.e. Rather, in all presented cases the exact energy functional minimization is achieved by using only a couple of simple mathematical tricks: 'completion of square' and integration by parts. A graphical representation of the Morse potential in equation (13). Reset your password. exact eigenfunctions in our proof, since they certainly exist and form It is well known that quantum mechanics can be formulated in an elegant and appealing way starting from variational first principles. Heisenberg's uncertainty principle is the essence of quantum mechanics. Theorem, which states that the energy of any trial wavefunction is For x > a, the wavefunction u(x) turns out to be, Since u(x) must be, together with its first derivative, continuous everywhere, it is necessary for the derivative of the sinusoidal function in equation (25) at x = a to be negative, thus implying, Moreover, since E0 is less than U0, from equation (27) it also follows that. After simple algebra we obtain. variational method (SVM), following the paper by two of the present authors [Phys. Schrödinger's equation for the stationary state u = u(x) reads. . i.e. In the final part of the paper (section 6) it will be shown how the procedure just described could be part of a possible elementary introduction to the so-called factorization method. VARIATIONAL METHODS IN RELATIVISTIC QUANTUM MECHANICS MARIA J. ESTEBAN, MATHIEU LEWIN, AND ERIC SER´ E´ Abstract. Riccardo Borghi https://orcid.org/0000-0002-4991-3156, Received 1 December 2017 The variational principles of classical mechanics differ from one another both by the form and by the manners of variation, and by their generality, but each principle, within the scope of its application, forms a unique foundation of and synthesizes, as it were, the entire mechanics … The variational method is the most powerful technique for doing working approximations when the Schroedinger eigenvalue equation cannot be solved exactly. Is the variational method useless if you already know the ground state energy? where  = h/2π, h being Planck's constant. Some hints aimed at guiding students to find the ground state of the Rosen-Morse potential are given in the appendix. Similarly to what was done for Morse's potential, to find the ground state of the Pöschl-Teller potential (30), the dimensionless parameter α defined in equation (15) is first introduced, i.e. analytically. In this way, the elementary character of the derivation will appear. Consider then equation (11), which will be recast in the following form: whose lhs can be interpreted in terms of the action of the differential operator x+{\rm{d}}/{\rm{d}}x on the ground state wavefunction u(x). Moreover, on using solely the Leibniz differentiation rule for the product, it is a trivial exercise to expand the operator in equation (53) as follows: so that, after substitution into equation (51), the Schrödinger equation for the harmonic oscillator takes on the factorized form. to find the optimum value . Consider then a harmonic oscillator with frequency ω, whose potential energy is. It is well known that quantum mechanics can be formulated in an elegant and appealing way starting from variational first principles. Variational Methods. The presence of the term \widehat{{{\boldsymbol{L}}}^{2}}/2{{mr}}^{2} into the Hamiltonian implies that the eigenvalues E will contain an amount of (positive) energy which has to be ascribed to the presence of centrifugal forces that tend to repel the electron from the force centre. © 2018 European Physical Society It can also be used to approximate the energies of a solvable system and then obtain the accuracy of the method by … Accordingly, such a direct connection could also be offered to more expert audiences (graduate students) who would benefit from the present derivation to better appreciate the elegance and powerfulness of the variational language. Consider then the potential profile sketched as a dashed line in figure 2, where the left barrier is supposed to be infinitely high. On expanding both sides of equation (A.2), it is not difficult to show that the parameters χ, β, and must satisfy the following algebraic relationships: Note that the first of the above equations coincides with equation (37). In other words, from equation (52) it is possible not only to retrieve the ground state wavefunction u(x), as it was done before, but also the corresponding value of the ground state energy. : To minimize the rhs of equation (7), the square in the numerator is first completed, which yields, then a partial integration is performed on the last integral. A pictorial representation of the Pöschl-Teller potential in equation (30). The variational method lies behind Hartree-Fock theory and the Students can be invited to check equation (68) for the entire catalogue presented here. Semiclassical approximation. Partial integration in both integrals into the numerator gives, from which it follows that the hydrogen ground energy is -{{ \mathcal E }}_{0}. configuration interaction method for the electronic structure of Figure 2. The variational method in quantum mechanics: an elementary. The first integral in the rhs of equation (33) is expanded as. Mechanics.In this study project, the Variational Principle has been applied to several scenarios, with the aim being to obtain an upper bound on the ground state energies of several quantum systems, for some of which, the Schrodinger equation cannot be easily solved. But there is more. Compared to perturbation theory, the variational As a matter of fact, it could result in being somewhat puzzling, for nonexpert students, to grasp why the oscillator zero-point energy value ω/2 should follow from the sole spatial localization. Nevertheless, in the present section we would offer teachers a way to introduce, again by using only elementary tools, a rather advanced topic of quantum mechanics, the so-called factorization method, introduced during the early days of quantum mechanics as a powerful algebraic method to solve stationary Schrödinger's equations [13–16]. Moreover, the key role played by particle localization is emphasized through the entire analysis. In the present paper a short catalogue of different celebrated potential distributions (both 1D and 3D), for which an exact and complete (energy and wavefunction) ground state determination can be achieved in an elementary … One of the most important byproducts of such an approach is the variational method. As a consequence, the number of quantum systems that can be adequately studied with a limited use of math is considerably small. To this end, let the integral be recast as follows: then search those values of χ and for equation (17) to be satisfied. They will be examined in section 3 and in section 4, respectively. Then also the stationary Schrödinger equation of the Morse oscillator, Students should be encouraged to prove that, starting from equation (38), the Schrödinger equation for the Pöschl-Teller potential (30) can also be factorized as. All (real) solutions of equation (1) describing bound energy's eigenstates must be square integrable on the whole real axis, The ground state for the potential U(x) can be found, in principle, without explicitly solving equation (1). This results from the Variational Variational principles. While this fact is evident for a particle in an infinite well (where the energy bound directly follows from boundary conditions), for the harmonic oscillator such a connection already turns out to be much less transparent. This would help to clarify how the minimization of the energy functional (5) can be carried out, in some fortunate cases, by using only 'completion of square' and integration by parts. You do not need to reset your password if you login via Athens or an Institutional login. of basis functions, such as. Such a small value would justify the above harmonic approximation of the ground state tone, which turns out to be about 4500 cm−1, in agreement with its experimental value [11]4 Variational methods, in particular the linear variational method, are the most widely used approximation techniques in quantum chemistry. Teaching quantum mechanics at an introductory (undergraduate) level is an ambitious but fundamental didactical mission. You are free to: • Share — copy or redistribute the material in any medium or format. which does coincide with equation (64) only if β(x) and satisfy the Riccati-type differential equation. but is still solvable). it is proportional to the well known radial function exp(−r/aB). In section 2 the 1D stationary Schrödinger equation and the variational method are briefly recalled, together with the main results of [2]. wavefunction can be written. In the present paper a short catalogue of different celebrated potential distributions (both 1D and 3D), for which an exact and complete (energy and wavefunction) ground state determination can be achieved in an elementary way, is illustrated. What has been shown so far is enough to cover at least two didactical units (lecture and recitation session). Some of them have been analyzed here. quantum mechanics. Remarkably, such a differential equation can easily be derived by using the variational approach used throughout the whole paper. This review is devoted to the study of stationary solutions of lin-ear and nonlinear equations from relativistic quantum mechanics, involving the Dirac operator. Schrödinger's equation, expressed via the above introduced 'natural units,' reads. These parameters are The variational principle Quantum mechanics 2 - Lecture 5 Igor Luka cevi c UJJS, Dept. Then, on inserting from equation (A.2) into equation (A.1) and on taking equation (A.3) into account, simple algebra gives. Before concluding the present section it is worth giving a simple but really important example of what kind of information could be, in some cases, obtained by only the ground state knowledge. The basis for this method is the variational principle. The second case we are going to deal with is the so-called Pöschl-Teller potential, defined as follows:5. The Rosen-Morse potential, originally proposed as a simple analytical model to study the energy levels of the NH3 molecule, can be viewed as a modification of the Pöschl-Teller potential in which the term -2\eta \tanh {kx} allows the asymptotic limits for x\to \pm \infty to split, as can be appreciated by looking at figure 5, where a pictorial representation of the potential (42) has been sketched. At the end of the functional minimization process, equation (21) has been obtained. A beautiful, didactically speaking, introduction to vibrational spectra of diatomic molecules can still be found on the Berkeley textbook [1]. One example of the variational method would be using the Gaussian Now partial integration is applied to the second integral in the numerator of equation (3), which transforms as follows: where use has been made of the spatial confinement condition in equation (2). International Conference on Variational Method, Variational Theory and Variational Principle in Quantum Mechanics scheduled on July 14-15, 2020 at Tokyo, Japan is for the researchers, scientists, scholars, engineers, academic, scientific and university practitioners to present research activities that might want to attend events, meetings, seminars, congresses, workshops, summit, and symposiums. Similarly as was done for the Pöschl-Teller, the integral into the numerator of equation (A.1) is written as a perfect square. variational method approximations to the exact wavefunction and No previous knowledge of calculus of variations is required. To this end, it is sufficient to multiply its left and right side by u and then integrate them over the whole real axis. approximate wavefunction and energy for the hydrogen atom would then The variational method was the key ingredient for achieving such a result. of the variational parameter , and then minimizing To find out more, see our, Browse more than 100 science journal titles, Read the very best research published in IOP journals, Read open access proceedings from science conferences worldwide, Quantum harmonic oscillator: an elementary derivation of the energy spectrum, Investigation of Bose-Einstein Condensates in, Generalized Morse potential: Symmetry and satellite potentials, Solutions to the Painlevé V equation through supersymmetric quantum mechanics, Quantum features of molecular interactions associated with time-dependent non-central potentials, A Laplace transform approach to the reflection and transmission of electrons at semi-infinite potential barriers, Two-year Postdoctoral/Temporary Scientist, Director of National Quantum Computing Centre. It is easy to prove that the same differential equation is also obtained by expanding the rhs of equation (62), thus completing our elementary proof. 39 035410. Accepted 16 February 2018 Interaction potential energy for the ground state of the hydrogen molecule as a function of the internuclear distance (dashed curve) [10], together with the fit provided by Morse's potential of equation (13) (solid curve). A fundamental three-dimensional (3D henceforth) problem, namely the determination of the hydrogen atom ground state, will also be presented in section 5. From equation (10) it also follows that, in order for the oscillator energy bound to be attained, the wavefunction must satisfy the following first order linear differential equation: whose general integral, that can be found with elementary tools (variable separation), is the well known Gaussian function. The need to avoid, as much as possible, the use of mathematical equipment that could not be still present within the toolbox of undergraduates necessarily limits the number of topics to be offered with an adequate level of detail. Published 13 April 2018, Riccardo Borghi 2018 Eur. In the present section, some hints are given to help students reaching the ground state of the potential into equation (42). (Refer Section 3 - Applications of the Variational Principle). Figure 5. variational method by obtaining the energy of as a function The variational method in quantum mechanics Gauss's principle of least constraint and Hertz's principle of least curvature Hilbert's action principle in general relativity, leading to the Einstein field equations . The variational method is the other main approximate method used in quantum mechanics. . To cite this article: Riccardo Borghi 2018 Eur. More often one is faced with a potential or a Hamiltonian for which exact methods are unavailable and approximate solutions must be found. BibTeX Variational method → Variational method (quantum mechanics) – I think that the move in 2009 was, unfortunately, a clear mistake. resulting trial wavefunction and its corresponding energy are It is natural to wonder whether the approach used in [2] is limited to the particularly simple mathematical structure of the harmonic oscillator potential or if it has a wider applicability. A possible elementary introduction to factorization could start again from the analysis of the harmonic oscillator potential recalled in section 2. we're applying the variational method to a problem we can't solve The work is organized in the form of a self-contained didactical unit. On the other hand, elementary derivations of Schrödinger's equation solutions constitute exceptions rather than the rule. The general solution of the factorization problem requires advanced mathematical techniques, like the use of a nonlinear differential equation. Variational principle, stationarity condition and Hückel method Variational approximate method: general formulation Let us consider asubspace E M of the full space of quantum states. It is useful to introduce 'natural units' for length and energy in order for the functional (5), as well as the corresponding Schrödinger equation, to be reduced to dimensionless forms. Now, similarly as done for the harmonic oscillator, consider the following differential operator: which, after expansion, takes on the form. Before proceeding to the minimization, it is better to recast equation (31) as follows: which implies that the energy must be greater than −1 (−U0 in physical units), as can be inferred from figure 4. This site uses cookies. This problem could be solved by the For radial functions the 3D integration reduces to a 1D integration. Its characterization is complete, as promised. of the quantum harmonic oscillator [2]. Moreover, from the above analysis it is also evident how the localization constraint in equation (2) is solely responsible for the above energy bound. It will be shown that the approach pursued throughout the present paper provides a didactically effective way to derive several examples of exact factorizations. Why would it make sense that the best approximate trial wavefunction Let the trial wavefunction be denoted In this way, the operator in equation (53) turns out to be Hermitian. In all above examples the minimization of the energy functional is achieved with the help of only two mathematical tricks: the so-called 'square completion' and the integration by parts, that should be part of the background of first-year Physics or Engineering students. The basis for this method is the variational principle. . It is well known that the study of quantum mechanics poses such challenging math problems which often may obscure the physics of the concepts to be developed. The variational principle Contents 1 Theory 2 The ground state of helium 3 The linear variational problem 4 Literature Revised 28 January 2018 of Physics, Osijek November 8, 2012 Igor Luka cevi c The variational principle. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. It should be pointed out how exact solutions of the Riccati equation (68) can be derived via a purely algebraic way, starting from a simple minimum principle.