# variational method in quantum mechanics

Teaching quantum mechanics at an introductory (undergraduate) level is an ambitious but fundamental didactical mission. The general solution of the factorization problem requires advanced mathematical techniques, like the use of a nonlinear differential equation. To this end, let the integral be recast as follows: then search those values of χ and for equation (17) to be satisfied. To cite this article: Riccardo Borghi 2018 Eur. Published 13 April 2018 • 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). 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. introduction. (Refer Section 3 - Applications of the Variational Principle). 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]. analytically. The final example we wish to offer is a simple and compact determination of the ground state of the hydrogen atom. . 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. variational method (SVM), following the paper by two of the present authors [Phys. Heisenberg's uncertainty principle is the essence of quantum mechanics. formally identical to the inequality in equation (24) once letting k ~ π/a. Compared to perturbation theory, the variational method can be more robust in situations where it is hard to determine a good unperturbed Hamiltonian (i.e., one which makes the … You are free to: • Share — copy or redistribute the material in any medium or format. The technique involves guessing a reason- with χ, of course, being the solution of equation (37). This allows calculating approximate wavefunctions such as molecular orbitals. What has been shown so far is enough to cover at least two didactical units (lecture and recitation session). 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. Why would it make sense that the best approximate trial wavefunction Ground State Energy of the Helium Atom by the Variational Method. 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. Schrödinger's equation for the electron wavefunction within the Coulomb electric field produced by the nucleus is first recalled. 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 … 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). 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]. 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. Variational Method. To this end, we shall let, and then search for the values of χ and such that equation (33) is fulfilled. The variational method is an approximate method used in quantum mechanics. The variational method is the other main approximate method used in quantum mechanics. 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). Number 3, 1 Dipartimento di Ingegneria, Università degli Studi 'Roma tre' Via Vito Volterra 62, I-00146 Rome, Italy. 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]. No previous knowledge of calculus of variations is required. A pictorial representation of the Rosen-Morse potential in equation (42). good unperturbed Hamiltonian, perturbation theory can be more In this way, the elementary character of the derivation will appear. resulting trial wavefunction and its corresponding energy are The true Morse oscillator energy lower bound is -{(1-\alpha /2)}^{2}. 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]. 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 variational method was the key ingredient for achieving such a result. Some hints aimed at guiding students to find the ground state of the Rosen-Morse potential are given in the appendix. One of the most important byproducts of such an approach is the variational method. Variational methods, in particular the linear variational method, are the most widely used approximation techniques in quantum chemistry. J. Phys. As usual, suitable units for length and energy are used to make the corresponding Schrödinger equation dimensionless. to find the optimum value . efficient than the variational method. 2. Figure 3. In other words, only radially symmetric wavefunctions, i.e. The variational method was the key ingredient for achieving such a result. In [2] it was shown that the energy functional in equation (5) can be minimized in an elementary way for the special case of the harmonic oscillator. The first integral into the rhs of equation (17) is expanded to have. 39 035410. This should help students to appreciate how some basic features of a phenomenon can sometimes be grasped even by using idealized, nonrealistic models. u = u(r), will be considered into equation (44). Fit parameters are U0  4.7 eV and k  2.0 Å−1. This allows calculating approximate wavefunctions such as molecular orbitals. 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. Frequently, the trial function is written as a linear combination On the other hand, in cases where there is a Figure 5. The celebrated Morse potential, described by the two-parameter function. 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). The variational method lies behind Hartree-Fock theory and the Some of them have been analyzed here. Such an unexpected connection is outlined in the final part of the paper. But there is more. 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). 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. exact eigenfunctions in our proof, since they certainly exist and form 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. Similar considerations hold for the Rosen-Morse potential. The integer M denotes the (ﬁnite) dimension of E M and fj Iig I=1;2;:::;M is a(not necessarily orthonormal)basis of that subspace. Revised 28 January 2018 but is still solvable). 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. We are not aware of previous attempts aimed at providing a variational route to factorization. The Variational Method. 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]. In quantum mechanics, the variational method is one way of finding approximations to the lowest energy eigenstate or ground state, and some excited states. Before continuing, it must be stressed once again how the above results have been obtained, after all, by imposing solely the localization constraint (2) on the energy functional (7). In the next section the same procedure will be used to find the ground state of the Morse oscillator. 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. Then, on evaluating the second integral in the numerator of equation (65) again by parts, i.e. Substitution from equation (4) into equation (3) gives. @article{osti_4783183, title = {A NEW VARIATIONAL PRINCIPLE IN QUANTUM MECHANICS}, author = {Newman, T J}, abstractNote = {Quantum theory is developed from a q-number (operator) action principle with a representation-invariant technique for limiting the number of independent system variables. Schrödinger's equation, expressed via the above introduced 'natural units,' reads. wavefunction for the problem, which consists of some adjustable : 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 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. 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. approximate wavefunction and energy for the hydrogen atom would then 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. 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. is the one with the lowest energy? 39 035410. You will only need to do this once. From equation (55), on again taking equation (52) into account, it follows that the energy of the ground state is just 1. To this end, the free parameters, χ, β, and are introduced, and their values are chosen in such a way that the following relation holds: with being a constant factor which contributes to the final expression of the ground state energy. As a further example, consider again the Morse potential of section 3. 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 variational method is the other main approximate method used in On the other hand, elementary derivations of Schrödinger's equation solutions constitute exceptions rather than the rule. This site uses cookies. 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. . Export citation and abstract 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. Functional minimization requires the knowledge of mathematical techniques that cannot be part of undergraduate backgrounds. Theorem, which states that the energy of any trial wavefunction is On coming back to physical units and on taking equation (15) into account, the ground energy is. where η ∈ (−1, 1). was proposed in 1929 by Morse [8] as a simple analytical model for describing the vibrational motion of diatomic molecules. In this way even graduate students could benefit from our elementary derivation to better appreciate the power and the elegance of the variational language. of basis functions, such as. 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. Moreover, from the above analysis it is also evident how the localization constraint in equation (2) is solely responsible for the above energy bound. Moreover, on using Bohr's radius aB = 2/me2 and the hydrogen ionization energy {{ \mathcal E }}_{0}={{me}}^{4}/2{{\hslash }}^{2} as unit length and unit energy, respectively, it is possible to recast equation (44) as follows: Similarly as was done for the 1D cases, we multiply both sides of equation (45) by u(r) and then integrate over the whole 3D space7 Similarly as was done for the Pöschl-Teller, the integral into the numerator of equation (A.1) is written as a perfect square. a complete set, even if we don't happen to know them. 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 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. If you have a user account, you will need to reset your password the next time you login. The basis for this method is the variational principle. Its characterization is complete, as promised. 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. The variational method is the most powerful technique for doing working approximations when the Schroedinger eigenvalue equation cannot be solved exactly. It is a useful analytical model to describe finite potential wells as well as anharmonic oscillators, and is sketched in figure 4. 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. VARIATIONAL METHODS IN RELATIVISTIC QUANTUM MECHANICS MARIA J. ESTEBAN, MATHIEU LEWIN, AND ERIC SER´ E´ Abstract. The solutions are found as critical points of an energy func-tional. To this end, it is sufficient to multiply its left and right side by u and then integrate them over the whole real axis. Schrödinger's equation for the stationary state u = u(x) reads. In this chapter, we will introduce two basic approaches—the variational and perturbation method. good unperturbed Hamiltonian (i.e., one which makes the perturbation small lengths and energies will again be measured in terms of U0 and α/k, respectively. be of the quantum harmonic oscillator [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. 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. Find out more. From equation (49) it also follows that the ground state wavefunction must be the solution of the differential equation. 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. Reset your password. 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. RIS. 2 To Franco Gori, on his eightieth birthday. 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 obtain the true energy lower bound, the square inside the integral into the numerator of equation (32) has to be completed. Compared to perturbation theory, the variational Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. that having the minimum energy, will be an eigenstate of \widehat{{{\boldsymbol{L}}}^{2}} corresponding to a null value of angular momentum. In this way, equation (5) takes on the following form2 They will be examined in section 3 and in section 4, respectively. 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. Is the variational method useless if you already know the ground state energy? where  = h/2π, h being Planck's constant. The present paper expounds a method which allows us to combine PT and the variation method in a single approach. View the article online for updates and enhancements. We know the ground state energy of the hydrogen atom is -1 Ryd, or -13.6 ev. To minimize the functional (16), the square into the integral in the numerator will first be completed. This makes our approach particularly suitable for undergraduates. This results from the Variational 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). In particular, on taking equation (2) into account, we have, so that, after simple algebra, equation (7) becomes [2]. of the variational parameter , and then minimizing 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. A possible elementary introduction to factorization could start again from the analysis of the harmonic oscillator potential recalled in section 2. The parameter a, that fixes the length scale, is expected to be proportional to k−1. A 'toy' model for the Morse potential. Of course, method can be more robust in situations where it's hard to determine a of Physics, Osijek November 8, 2012 Igor Luka cevi c The variational principle. Variational methods in quantum mechanics are customarily presented as invaluable techniques to find approximate estimates of ground state energies. variational method approximations to the exact wavefunction and By continuing to use this site you agree to our use of cookies. we're applying the variational method to a problem we can't solve For this reason the ground state, i.e. For radial functions the 3D integration reduces to a 1D integration. In this approach, the origin of the nite minimum value of uncertainty is attributed to the non-di erentiable (virtual) trajectory of a quantum particle and then both of the Kennard and Robertson-Schr odinger inequalities in The first integral in the rhs of equation (33) is expanded as. Note that the first term in equation (29) does coincide with the ground state energy of the harmonic approximation of the Morse potential (13), as can be easily proved by taking the second derivative of the potential at x = 0. Moreover, on further letting x\to \alpha x, after simple algebra equation (14) can be recast as follows: Figure 1. Remarkably, such a differential equation can easily be derived by using the variational approach used throughout the whole paper. Let the trial wavefunction be denoted parameters called variational parameters.'' Nevertheless, there also exist many problems that may not be solved classically even with the clas-sical variational method [17{19]. Variational principles. Lett. combination of the exact eigenfunctions . 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. 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. For the harmonic potential two natural units are the quantities \sqrt{{\hslash }/m\omega } and ω/2 for length and energy, respectively. Nevertheless, that doesn't prevent us from using the 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. 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. This wave function contains a lot more information than just the ground state energy. The variational method winds up giving you a wave function that is supposed to approximate the ground state wave function. It could also be worth exploring the Infeld/Hull catalogue to find, and certainly there are, other interesting cases to study. Any trial function can formally be expanded as a linear You will only need to do this once. 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. The second case we are going to deal with is the so-called Pöschl-Teller potential, defined as follows:5. The Variational Method 1. Published 13 April 2018, Riccardo Borghi 2018 Eur. The knowledge of higher-order eigenstates would require mathematical techniques that are out of the limits and the scopes of the present paper. 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. After simple algebra the corresponding energy functional is then obtained, where it will now be assumed henceforth that the limits of r-integrals are [0,\infty ). No. The variational method in quantum mechanics: an elementary. The variational method was the key ingredient for achieving such a result. . hydrogen atom ground state. 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 well known that quantum mechanics can be formulated in an elegant and appealing 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 main result found in [2] will now be briefly resumed. Naturally, many other exist … 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. 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. It is well known that quantum mechanics can be formulated in an elegant and appealing way starting from variational first principles. 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. 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. In this way it is easy to prove that equation (5) reduces to. 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). Two of these potentials are one-dimensional (1D henceforth), precisely the Morse and the Pöschl-Teller potentials. 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. There exist only a handful of problems in quantum mechanics which can be solved exactly. This is because there exist highly entangled many-body states that configuration interaction method for the electronic structure of Variational Methods. From: Elementary Molecular Quantum Mechanics (Second Edition), 2013. function wavefunction can be written. In this way, the operator in equation (53) turns out to be Hermitian. In section 2 the 1D stationary Schrödinger equation and the variational method are briefly recalled, together with the main results of [2]. 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. To avoid symbol proliferation, the same notations will be used to denote physical as well as dimensionless quantities. One of the most important byproducts of such an approach is the variational method. It appears that quantities k−1 and U0 provide natural units for length and energy, respectively. 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. The basic idea of the variational method is to guess a trial'' This can be proven easily. 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). At the end of the functional minimization process, equation (21) has been obtained. However, it was pointed out how such inclusion does not dramatically alter the resulting vibrational spectrum [8]. Variational Methods The variational technique represents a completely diﬀerent way of getting approximate energies and wave functions for quantum mechanical systems. Now, similarly as done for the harmonic oscillator, consider the following differential operator: which, after expansion, takes on the form. © 2018 European Physical Society Students should be encouraged to study, for instance, the so-called Rosen-Morse potential, defined by [12]. Semiclassical approximation. 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.