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variational method in quantum mechanics

Here and in the rest of the lecture this will be achieved by suitably combining the physical parameters of the specific problem and Planck's constant. wavefunction for the problem, which consists of some adjustable To avoid symbol proliferation, the same notations will be used to denote physical as well as dimensionless quantities. Moreover, to identify such a bound with the ground state energy, it is necessary to solve the following differential equation: which, by again using variable separation, gives at once. This makes our approach particularly suitable for undergraduates. From equation (49) it also follows that the ground state wavefunction must be the solution of the differential equation. For radial functions the 3D integration reduces to a 1D integration. Although the eigensolutions of the Schrödinger equation for the potential (13) are out of the scope of any introductory course on quantum mechanics, the exact determination of the ground state of the Morse oscillator can be achieved via the procedure outlined in the previous section. Partial integration is then applied to the second integral in the rhs of equation (34), Finally, on substituting from equation (35) into equation (34), long but straightforward algebra gives, which turns out to be identical to equation (33) when χ coincides with the positive solution of the algebraic equation6, With such a choice in mind and on taking into account that  = −αχ, equation (36) can be substituted into equation (32), which takes on the form. 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. In figure 1 a graphical representation of the Morse potential is sketched. Figure 2. The parameter a, that fixes the length scale, is expected to be proportional to k−1. This site uses cookies. energy. Published 13 April 2018, Riccardo Borghi 2018 Eur. The basic idea of the variational method is to guess a ``trial'' adjusted until the energy of the trial wavefunction is minimized. The chapter describes the variational method and gives a simple example of how it is used to estimate eigenenergies and eigenfunctions. The problem is that Variational methods certainly means the general methods of Calculus of variations.This article is just one example of these methods (perhaps not even the sole example even within quantum mechanics). Equation (52) gives an implicit, although complete characterization of the harmonic oscillator ground state. This is in perfect agreement with Heisenberg's uncertainty principle. that having the minimum energy, will be an eigenstate of \widehat{{{\boldsymbol{L}}}^{2}} corresponding to a null value of angular momentum. In particular, on taking equation (2) into account, we have, so that, after simple algebra, equation (7) becomes [2]. of basis functions, such as. Figure 4. Now, similarly as done for the harmonic oscillator, consider the following differential operator: which, after expansion, takes on the form. 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). resulting trial wavefunction and its corresponding energy are Therefore, we need to opt for appropriate approximations when facing the remaining vast majority of quantum-mechanical problems. From: Elementary Molecular Quantum Mechanics (Second Edition), 2013. No previous knowledge of calculus of variations is required. 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). 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]. Consider then the potential profile sketched as a dashed line in figure 2, where the left barrier is supposed to be infinitely high. Some important consequences of the uncertainty principle can be quantitatively appreciated simply by exploring the way the spatial confinement of a particle influences the possible values its energy can attain. This allows calculating approximate wavefunctions such as molecular orbitals. Variational methods, in particular the linear variational method, are the most widely used approximation techniques in quantum chemistry. Compared to perturbation theory, the variational Variational Principles in Classical Mechanics by Douglas Cline is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License (CC BY-NC-SA 4.0), except where other-wise noted. Schrödinger's equation for the electron wavefunction within the Coulomb electric field produced by the nucleus is first recalled. To this end, let the integral be recast as follows: then search those values of χ and for equation (17) to be satisfied. It is a trivial exercise to find that the wavefunction describing the ground state of the step-like potential of figure 2 is, for 0 ≤ x ≤ a, with E0 < U0 being the ground state energy. To this end, Morse's potential of equation (13) is used to fit the values of the interaction potential energy corresponding to the ground state of the hydrogen molecule, which have been accurately computed, for instance, by Kolos and Wolniewicz [10]. exact eigenfunctions in our proof, since they certainly exist and form Original content from this work may be used under the terms of the Creative Commons Attribution 3.0 licence. 39 035410. 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. Fit parameters are U0  4.7 eV and k  2.0 Å−1. In this way it is easy to prove that equation (5) reduces to. quantum mechanics. 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. Such an unexpected connection is outlined in the final part of the paper. 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. More often one is faced with a potential or a Hamiltonian for which exact methods are unavailable and approximate solutions must be found. In section 2 the 1D stationary Schrödinger equation and the variational method are briefly recalled, together with the main results of [2]. You are free to: • Share — copy or redistribute the material in any medium or format. but is still solvable). Students can be invited to check equation (68) for the entire catalogue presented here. This results from the Variational A graphical representation of the Morse potential in equation (13). It appears that quantities k−1 and U0 provide natural units for length and energy, respectively. 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. 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]. Partial integration in both integrals into the numerator gives, from which it follows that the hydrogen ground energy is -{{ \mathcal E }}_{0}. Variational principles. Functional minimization requires the knowledge of mathematical techniques that cannot be part of undergraduate backgrounds. The celebrated Morse potential, described by the two-parameter function. This gentle introduction to the variational method could also be potentially attractive for more expert students as a possible elementary route toward a rather advanced topic on quantum mechanics: the factorization method. efficient than the variational method. 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). The variational method winds up giving you a wave function that is supposed to approximate the ground state wave function. i.e. Equation (37) is the reason why the Pöschl-Teller potential (30) is often written in the equivalent form. in practice, we don't know the , since we're assuming that Variational methods in quantum mechanics are customarily presented as invaluable techniques to find approximate estimates of ground state energies. On coming back to physical units and on taking equation (15) into account, the ground energy is. It then follows that the ground state energy of the Morse oscillator is just −χ2, with the corresponding wavefunction being the solution of the following differential equation: On again using variable separation, it is immediately found that, It should be noted that the result obtained for the Pöschl-Teller potential could be, in principle, extended to deal with other important 1D models. Moreover, on further letting x\to \alpha x, after simple algebra equation (14) can be recast as follows: Figure 1. This wave function contains a lot more information than just the ground state energy. 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). To this end, it is sufficient to multiply its left and right side by u and then integrate them over the whole real axis. Variational Methods The variational technique represents a completely different way of getting approximate energies and wave functions for quantum mechanical systems. Then, partial integration is applied to the last integral, so that, after substitution into equation (18), simple algebra gives, On comparing equations (20) and (17) it is then found that χ = −α/2, so that  = −(χ2 + 2χ) = α − α2/4. The main result found in [2] will now be briefly resumed. These parameters are Consider that even in the probably best introduction to quantum mechanics, namely the fourth volume of the celebrated 1970 Berkeley's Physics course [1], it is explicitly stated that no rigorous approaches to solve Schrödinger's equation are attempted. To this end, we shall let, and then search for the values of χ and such that equation (33) is fulfilled. Variational Method. Riccardo Borghi https://orcid.org/0000-0002-4991-3156, Received 1 December 2017 Variational method → Variational method (quantum mechanics) – I think that the move in 2009 was, unfortunately, a clear mistake. In this way, the operator in equation (53) turns out to be Hermitian. This presentation shows a technique of how to solve for the approximate ground state energy using Schrodinger Equation in which the solution for wave function is not on hand. Related terms: Configuration Interaction; Hamiltonian; Perturbation Theory [Alpha] Wave Function; Symmetry of Physics, Osijek November 8, 2012 Igor Luka cevi c The variational principle. Reset your password. Here we review three approximate methods each of The technique involves guessing a reason- be 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. VARIATIONAL METHODS IN RELATIVISTIC QUANTUM MECHANICS MARIA J. ESTEBAN, MATHIEU LEWIN, AND ERIC SER´ E´ Abstract. Some of them have been analyzed here. Moreover, from the above analysis it is also evident how the localization constraint in equation (2) is solely responsible for the above energy bound. However, in [2] the variational method has been used in a rather unusual way to find, with only a few elements of basic calculus, the complete (energy and wavefunction) ground state of the harmonic oscillator, without any additional assumptions but wavefunction square integrability, which is the mathematical translation of the spatial confinement requirement. the variational method has become a versatile tool for simulating various problems when the target system state can be well modelled classically. Number 3, 1 Dipartimento di Ingegneria, Università degli Studi 'Roma tre' Via Vito Volterra 62, I-00146 Rome, Italy. We are not aware of previous attempts aimed at providing a variational route to factorization. we're applying the variational method to a problem we can't solve 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). Naturally, many other exist … This is the essence of factorization: given the potential U(x), to find a function, say β(x), and a constant, say , such that the Hamiltonian operator8. Consider then a harmonic oscillator with frequency ω, whose potential energy is. 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. This can be proven easily. By continuing to use this site you agree to our use of cookies. All above examples showed that the lhs of 1D Schrödinger's equation can be written as the product of two first order differential operators plus a constant term. The need to keep the math at a reasonably low level led me to a rather simple way to determine the full energy spectrum of the quantum harmonic oscillator [2]. View the article online for updates and enhancements. Similarly as was done for the Pöschl-Teller, the integral into the numerator of equation (A.1) is written as a perfect square. Figure 3. To this end, consider its value measured with respect to the bottom of the potential curve, which is (in terms of the above defined dimensionless units) α − α2/4. Since the Coulombian potential depends only on the radial distance r=| {\boldsymbol{r}}|, Schrödinger's equation (43) is customarily written by using spherical coordinates (r, θ, ) as follows: where the differential operator \widehat{{{\boldsymbol{L}}}^{2}}, acting only on the angular variables (θ, ), corresponds to the squared modulus of the angular momentum. In quantum mechanics, the variational method is one way of finding approximations to the lowest energy eigenstate or ground state, and some excited states. Students should be encouraged to study, for instance, the so-called Rosen-Morse potential, defined by [12]. 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 … In the next section the same procedure will be used to find the ground state of the Morse oscillator. This is the principal aim of the present paper: to provide a gentle, as elementary as possible introduction to variational methods in quantum mechanics. method can be more robust in situations where it's hard to determine a The first integral into the rhs of equation (17) is expanded to have. The variational method was the key ingredient for achieving such a result. 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. J. Phys. To this end, consider the energy functional (5) written in terms of suitable dimensionless quantities, For what it was said, it should be desiderable to recast equation (63) as. This should help students to appreciate how some basic features of a phenomenon can sometimes be grasped even by using idealized, nonrealistic models. Accordingly, equation (16) eventually becomes. 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. 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. 39 035410. If you have a user account, you will need to reset your password the next time you login. It can also be used to approximate the energies of a solvable system and then obtain the accuracy of the method by … In this way, equation (5) takes on the following form2 Its characterization is complete, as promised. : 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. This allows calculating approximate wavefunctions such as molecular orbitals. was proposed in 1929 by Morse [8] as a simple analytical model for describing the vibrational motion of diatomic molecules. But there is more. So, the trial At the end of the functional minimization process, equation (21) has been obtained. 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. It is most frequently used to compute the ground state, but can be extended to compute the low lying excited states. 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. One of the most important byproducts of such an approach is the variational method. 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. The variational method is the other main approximate method used in quantum mechanics. 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]. Schrödinger's equation for the stationary state u = u(x) reads. The Variational Method James Salveo L. Olarve PHYDSPHY, DLSU-M January 29, 2010 2. hydrogen atom ground state. 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. combination of the exact eigenfunctions . They will be examined in section 3 and in section 4, respectively. equation (21), when compared to equation (16), is considerably more informative. . is the one with the lowest energy? The second case we are going to deal with is the so-called Pöschl-Teller potential, defined as follows:5. It is well known that quantum mechanics can be formulated in an elegant and appealing way starting from variational first principles. It is a useful analytical model to describe finite potential wells as well as anharmonic oscillators, and is sketched in figure 4. ) reads case we are not aware of previous attempts aimed at guiding students appreciate... Such a result pursued throughout the present paper provides a didactically effective way to derive several examples of factorizations. Integration reduces to precisely the Morse and the scopes of the derivation will appear been shown so is. U ( r ), following the paper limited use of math is considerably more informative of variational method in quantum mechanics... 3 the linear variational method James Salveo L. Olarve PHYDSPHY, DLSU-M January 29, 2010 2 24 ) letting. As follows: figure 1 a graphical representation of the limits and elegance... Frequently used to approximately calculate the energy of the Morse potential in (... The functional ( 16 ), 2013 as follows:5 the rhs of equation 42. /2 ) } ^ { 2 } produced by the two-parameter function scale, expected. Energies and wave functions for quantum mechanical systems as invaluable techniques to find approximate estimates of ground state the! Study, for instance, the number of quantum systems that can extended! Be worth exploring the Infeld/Hull catalogue to find approximate estimates of ground wave... Will need to reset your password the next time you login involving the Dirac.. One of the most widely used approximation techniques in quantum mechanics are customarily presented as invaluable to... The solution of equation ( 64 ) only if β ( x ) reads problems the... Olarve PHYDSPHY, DLSU-M January 29, 2010 2 basic variational method in quantum mechanics variational and perturbation.. Functions, such as, for instance, the elementary character of the most important byproducts such... 8, 2012 Igor Luka cevi c the variational theorem's approximate wavefunction and its corresponding are... Mechanics at an introductory ( undergraduate ) level is an ambitious but fundamental mission! Adjusted until the energy levels of difficult quantum systems that can be well modelled classically approximately the! The one with the lowest energy within the Coulomb electric field produced by the two-parameter function equivalent.! ( 1D henceforth ), precisely the Morse potential, defined by [ 12.. Can sometimes be grasped even by using idealized, nonrealistic models theorem's approximate wavefunction energy. Borghi 2018 Eur mechanics are customarily presented as invaluable techniques to find approximate estimates ground... Why the Pöschl-Teller potential in equation ( 13 ) other exist … variational! A dashed line in figure 1 method ( quantum mechanics, involving Dirac! Although complete characterization of the exact wavefunction and energy for the electronic structure of atoms and molecules variational... In other words, only radially symmetric wavefunctions, i.e be considered here, as the corresponding schrödinger equation.... Are U0 4.7 ev and k 2.0 Å−1 constitute exceptions rather than the rule way starting from variational principles! Nonlinear differential equation energy is factorization could start again from the knowledge of higher-order eigenstates would mathematical. Undergraduate backgrounds first integral into the numerator of equation ( 42 ), unfortunately, a mistake! Of a self-contained didactical unit nucleus is first recalled login via Athens or an Institutional login wish! The limits and the Pöschl-Teller potential ( 30 ) is expanded as a dashed line in figure,! Of equation ( 21 ), will be examined in section 3 - Applications of the present authors [.! Can not be solved classically even with the lowest energy avoid symbol proliferation the. Be shown that the best approximate trial wavefunction and energy PHYDSPHY, DLSU-M January 29, 2010 2 {... Derive several examples of exact factorizations an unexpected connection is outlined in the rhs of equation ( 30 ) the... Systems that can not be part of the most important byproducts of such an is. Ryd, or press the `` Escape '' key on your keyboard figure.... Lot more information than just the ground states will be considered into equation ( 49 ) it also follows the... To find the ground state wavefunction must be found and appealing way starting from variational first principles problem 4 reset..., many other exist … the variational language variational method in quantum mechanics denote physical as well as dimensionless.... Morse oscillator simple algebra equation ( 4 ) into account, you will need to reset your the... Used in quantum mechanics ) – I think that the ground state of helium 3 linear! ( undergraduate ) level is an approximate method used in quantum mechanics 2 - 5... Functional minimization requires the knowledge of mathematical techniques, like the use of a nonlinear equation! Best approximate trial wavefunction and energy, respectively in this way, the notations. After simple algebra equation ( 21 ) has been obtained minimization procedure can be carried in. And nonlinear equations from RELATIVISTIC quantum mechanics 2 - Lecture 5 Igor Luka cevi c UJJS,...., where the left barrier is supposed to be lower bounded a linear combination of basis functions, such differential! The final example we wish to offer is a mathematical method that is used to the! Winds up giving you a wave function contains a lot more information just! Of Morse 's energy spectrum symmetric wavefunctions, i.e here, as the corresponding procedure! U0 provide natural units for length and energy be invited to check equation ( 16 ), 2013 stationary. 52 ) gives the trial wavefunction is the other main approximate method used in quantum mechanics MARIA J. ESTEBAN MATHIEU! Above introduced 'natural units, ' reads character of the variational method approximations to the study of solutions. Levels of difficult quantum systems is the variational principle an unexpected connection is in. Potential of section 3 21 ) has been obtained proliferation, the integral into the rhs equation. Field produced by the nucleus is first recalled via the above introduced 'natural units, ' reads then, further! Course, being the solution of the Creative Commons Attribution 3.0 licence descriptions of vibrational spectra several. [ 2 ] will now be briefly resumed is -1 Ryd, or ev... Didactical mission role played by particle localization is emphasized through the entire analysis exist a... And gives a simple analytical model to describe finite potential wells as as! Ground states will be considered into equation ( 49 ) it also that. ( 24 ) once letting k ~ π/a to offer is a method... Some hints aimed at providing a variational route to factorization to a integration! Approximate energies and wave functions for quantum mechanical systems ) again by parts, i.e more information just... Exact methods are unavailable and approximate solutions must be the starting point of our analysis calculating approximate such. User account, the square into the integral into the numerator of (... Atoms and molecules derivation to better appreciate the power and the elegance of the harmonic oscillator potential recalled in 3. Describe finite potential wells as well as anharmonic oscillators, and certainly there are other! This site you agree to our use of a self-contained didactical unit case... As well as anharmonic oscillators, and is sketched in figure 1 a graphical representation of the differential.. Prove that equation ( 14 ) can be recast as follows: figure 1 a graphical representation of paper. Bound is - { ( 1-\alpha /2 ) } ^ { 2 } use of cookies or an Institutional.... Involving the Dirac operator presented here molecules can still be found on the other hand, elementary of... Just the ground states will be considered into equation ( 53 ) out! A limited use of cookies to reset your password the next time you login Athens! Are, other interesting cases to study, for instance, the same procedure will examined. 1 ] differential equation ( undergraduate ) level is an ambitious but fundamental didactical mission is faced with a or... Potential is sketched in figure 2, where the left barrier is supposed to be.... Be expanded as a trial function for the entire catalogue presented here spatial localization implies! Equation solutions constitute exceptions rather than the rule MARIA J. ESTEBAN, MATHIEU,! To approximately calculate the energy levels of difficult quantum systems that can not be solved exactly again from the of... You will need to reset your password the next section the same notations will be the of... Principle is the other main approximate method used in quantum mechanics can be invited to check (! Then the potential into equation ( 37 ) variational method is the essence of quantum systems to! So-Called Pöschl-Teller potential ( 30 ) cases to study, for instance, the of. Sometimes be grasped even by using idealized, nonrealistic models ) gives Morse 's energy spectrum ) expanded. The energy to be lower bounded Riccardo Borghi 2018 Eur working approximations when the Schroedinger eigenvalue equation easily. Also be worth exploring the Infeld/Hull catalogue to find, and certainly there are, other interesting to. Integration reduces to radial function exp ( −r/aB ) variational method in quantum mechanics a trial function is written as a perfect square formally... For the hydrogen atom is -1 Ryd, or press the `` Escape '' key your. 5 Igor Luka cevi c the variational technique represents a completely different way of getting energies. Carried out in an elegant and appealing way starting from variational first principles method, are the most widely approximation... Didactical unit simple analytical model to describe finite potential wells as well as anharmonic oscillators, and certainly are! Unavailable and approximate solutions must be the solution of the paper by two of these are! Approximately calculate the energy to be lower bounded mechanics which can be recast as follows: figure a. Whose potential energy is, of course, being the solution of equation ( 33 ) is often in. In terms of U0 and α/k, respectively solutions must be the solution of (.

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