The present paper expounds a method which allows us to combine PT and the variation method in a single approach. 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. On the other hand, in cases where there is a Nevertheless, there also exist many problems that may not be solved classically even with the clas-sical variational method [17{19]. It is well known that quantum mechanics can be formulated in an elegant and appealing way starting from variational first principles. wavefunction for the problem, which consists of some adjustable In quantum mechanics, the variational method is one way of finding approximations to the lowest energy eigenstate or ground state, and some excited states. Variational methods in quantum mechanics are customarily presented as invaluable techniques to find approximate estimates of ground state energies. To avoid symbol proliferation, the same notations will be used to denote physical as well as dimensionless quantities. Functional minimization requires the knowledge of mathematical techniques that cannot be part of undergraduate backgrounds. 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. 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. where η ∈ (−1, 1). As a consequence, the number of quantum systems that can be adequately studied with a limited use of math is considerably small. This allows calculating approximate wavefunctions such as molecular orbitals. 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). Frequently, the trial function is written as a linear combination 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]. 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. The second case we are going to deal with is the so-called Pöschl-Teller potential, defined as follows: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. 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). with χ, of course, being the solution of equation (37). 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 is because there exist highly entangled many-body states that A graphical representation of the Morse potential in equation (13). The variational method was the key ingredient for achieving such a result. where it will be tacitly assumed henceforth that any integration has to be carried out across the whole real axis (-\infty ,+\infty ). So, the trial Teaching quantum mechanics at an introductory (undergraduate) level is an ambitious but fundamental didactical mission. 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. The variational method in quantum mechanics: an elementary. analytically. This is shown in figure 3, where the dashed curve contains the KW values, while the solid curve gives the behaviour of Morse's potential for U0  4.7 eV and k  2.0 Å−1. 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. where  = h/2π, h being Planck's constant. It can also be used to approximate the energies of a solvable system and then obtain the accuracy of the method by … 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. Then, on evaluating the second integral in the numerator of equation (65) again by parts, i.e. 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 solutions are found as critical points of an energy func-tional. The Variational Method 1. The variational method is the most powerful technique for doing working approximations when the Schroedinger eigenvalue equation cannot be solved exactly. J. Phys. It could also be worth exploring the Infeld/Hull catalogue to find, and certainly there are, other interesting cases to study. 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. Then, on identifying m = mp/2 as the reduced mass of the H2 molecules, equation (13) gives at once α  0.06. Figure 2. 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. equation (21), when compared to equation (16), is considerably more informative. efficient than the variational method. The variational principle Contents 1 Theory 2 The ground state of helium 3 The linear variational problem 4 Literature Let the trial wavefunction be denoted In other words, only radially symmetric wavefunctions, i.e. 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. Therefore, we need to opt for appropriate approximations when facing the remaining vast majority of quantum-mechanical problems. Number 3, 1 Dipartimento di Ingegneria, Università degli Studi 'Roma tre' Via Vito Volterra 62, I-00146 Rome, Italy. Consider then the potential profile sketched as a dashed line in figure 2, where the left barrier is supposed to be infinitely high. Similarly as was done for the Pöschl-Teller, the integral into the numerator of equation (A.1) is written as a perfect square. Compared to perturbation theory, the variational On the other hand, elementary derivations of Schrödinger's equation solutions constitute exceptions rather than the rule. 39 035410. 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. . A 'toy' model for the Morse potential. Figure 3. Equation (5) will be the starting point of our analysis. 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. In this way even graduate students could benefit from our elementary derivation to better appreciate the power and the elegance of the variational language. In this way it is easy to prove that equation (5) reduces to. As a further example, consider again the Morse potential of section 3. Theorem, which states that the energy of any trial wavefunction is To obtain the true energy lower bound, the square inside the integral into the numerator of equation (32) has to be completed. was proposed in 1929 by Morse [8] as a simple analytical model for describing the vibrational motion of diatomic molecules. A pictorial representation of the Rosen-Morse potential in equation (42). Its characterization is complete, as promised. a complete set, even if we don't happen to know them. Variational Method. quantum mechanics. The variational method was the key ingredient for achieving such a result. If you have a user account, you will need to reset your password the next time you login. To this end, it is sufficient to multiply its left and right side by u and then integrate them over the whole real axis. The final example we wish to offer is a simple and compact determination of the ground state of the hydrogen atom. Schrödinger's equation, expressed via the above introduced 'natural units,' reads. the variational method has become a versatile tool for simulating various problems when the target system state can be well modelled classically. It is well known that quantum mechanics can be formulated in an elegant and appealing To this end, we shall let, and then search for the values of χ and such that equation (33) is fulfilled. Moreover, from the above analysis it is also evident how the localization constraint in equation (2) is solely responsible for the above energy bound. Ground State Energy of the Helium Atom by the Variational Method. 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. The Variational Method. Related terms: Configuration Interaction; Hamiltonian; Perturbation Theory [Alpha] Wave Function; Symmetry European Journal of Physics, In quantum mechanics, the variational method is one way of finding approximations to the lowest energy eigenstate or ground state, and some excited states. of the variational parameter , and then minimizing A possible elementary introduction to factorization could start again from the analysis of the harmonic oscillator potential recalled in section 2. formally identical to the inequality in equation (24) once letting k ~ π/a. @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. Only the ground states will be considered here, as the corresponding minimization procedure can be carried out in an elementary way. The variational method is the other main approximate method used in quantum mechanics. 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. Published 13 April 2018, Riccardo Borghi 2018 Eur. 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. One of the most important byproducts of such an approach is the variational method. This review is devoted to the study of stationary solutions of lin-ear and nonlinear equations from relativistic quantum mechanics, involving the Dirac operator. It will be shown that the approach pursued throughout the present paper provides a didactically effective way to derive several examples of exact factorizations. Similar considerations hold for the Rosen-Morse potential. 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. RIS. of Physics, Osijek November 8, 2012 Igor Luka cevi c The variational principle. Some hints aimed at guiding students to find the ground state of the Rosen-Morse potential are given in the appendix. 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 theorem's One of the most important byproducts of such an approach is the variational method. J. Phys. However, it was pointed out how such inclusion does not dramatically alter the resulting vibrational spectrum [8]. A382, 1472 (2018)]. Figure 4. Then, on inserting from equation (A.2) into equation (A.1) and on taking equation (A.3) into account, simple algebra gives. 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. We are not aware of previous attempts aimed at providing a variational route to factorization. You will only need to do this once. 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. Schrödinger's equation for the electron wavefunction within the Coulomb electric field produced by the nucleus is first recalled. variational method by obtaining the energy of as a function This site uses cookies. Click here to close this overlay, or press the "Escape" key on your keyboard. 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 ). . As usual, suitable units for length and energy are used to make the corresponding Schrödinger equation dimensionless. Semiclassical approximation. During the minimization process, the starting energy functional is reduced to the sum of two terms: the first is the energy bound, the second is a nonnegative functional that vanishes when the wavefunction is the solution of a suitable first order linear ordinary differential equation, which can be solved again with elementary math tools (variable separation). To this end, let the integral be recast as follows: then search those values of χ and for equation (17) to be satisfied. Of course, The 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. The basic idea of the variational method is to guess a ``trial'' At the end of the functional minimization process, equation (21) has been obtained. Variational principles. i.e. The variational method is the other main approximate method used in In particular, on taking equation (2) into account, we have, so that, after simple algebra, equation (7) becomes [2]. we're applying the variational method to a problem we can't solve Reset your password. to find the optimum value . Such an unexpected connection is outlined in the final part of the paper. parameters called ``variational parameters.'' But there is more. 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. lengths and energies will again be measured in terms of U0 and α/k, respectively. This makes our approach particularly suitable for undergraduates. 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 practice, we don't know the , since we're assuming that Revised 28 January 2018 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. 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. In this chapter, we will introduce two basic approaches—the variational and perturbation method. Variational method → Variational method (quantum mechanics) – I think that the move in 2009 was, unfortunately, a clear mistake. 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). good unperturbed Hamiltonian, perturbation theory can be more On coming back to physical units and on taking equation (15) into account, the ground energy is. 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. 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]. Partial integration in both integrals into the numerator gives, from which it follows that the hydrogen ground energy is -{{ \mathcal E }}_{0}. 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. 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]. 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. The parameter a, that fixes the length scale, is expected to be proportional to k−1. 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 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. From equation (10) it follows that the oscillator energy cannot assume values less than ω/2 (when expressed through physical units). 2. No. method can be more robust in situations where it's hard to determine a Remarkably, such a differential equation can easily be derived by using the variational approach used throughout the whole paper. From equation (49) it also follows that the ground state wavefunction must be the solution of the differential equation. 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). energy. 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. It is a useful analytical model to describe finite potential wells as well as anharmonic oscillators, and is sketched in figure 4. Heisenberg's uncertainty principle is the essence of quantum mechanics. © 2018 European Physical Society In particular, spatial localization unavoidably implies the energy to be lower bounded. 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). 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. Original content from this work may be used under the terms of the Creative Commons Attribution 3.0 licence. 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