Variational Principle Quantum

7 Key excerpts on "Variational Principle Quantum"

• 

  eBook - PDF

  Variational Calculations In Quantum Field Theory: Proceedings Of The International Workshop

  • L Polley, D E L Pottinger(Authors)
  • 1988(Publication Date)
  • World Scientific

    (Publisher)

  Difficulties in Applying the Variational Principle to Quantum Field Theories 1 Richard P. Feynman California Institute of Technology Pasadena, California 91125, U.S.A. Introduction I'd like to talk on some work I did on the variational principle in field theory. At one time I thought that the brute force method of doing arithmetic on the machines will never get anywhere and we will probably end with something more old-fashioned, i.e. some analysis plus the machines to help us with the analytic equations, and so I tried to do something along these lines with quantum chromodynamics. So I'm talking on the subject of the application of the variational principle to field theoretic problems, but in particular to quantum chromodynamics. I'm going to give away what I want to say, which is that I didn't get anywhere! I got very discouraged and I think I can see why the variational principle is not very useful. So I want to take, for the sake of argument, a very strong view -which is stronger than I really believe - and argue that it is no damn good at all! Let us review why the variational principle has gotten a good reputation. Let's say you apply it to something like atoms or to simple problems with a small number of variables, using the usual analytic methods to get a quantity called the total energy, a quantity which is of direct physical significance. The energy levels of atoms are very interesting, measurable quantities and they can be calculated with accuracy. It was noted that if one had a wave function which had some measure of error, say 10 percent, then the error in the energy would be of order 1 percent. The error in the energy is quadratic in the error in the wave function. So, by not having a perfect wave function, you can still get very good values for the energy and that's why the variational method has gotten a good reputation. But it has never been a powerful way of getting, with accuracy, the wave function itself.

  Sign up to read

  Learn more about book
• 

  eBook - ePub

  The Quantum Mechanics of Many-Body Systems

  Second Edition

  • D.J. Thouless(Author)
  • 2013(Publication Date)
  • Dover Publications

    (Publisher)

  III. Variational Methods

  1. The Hartree–Fock equations

  Schrödinger’s variational principle in quantum mechanics tells us that the expectation value of the Hamiltonian for an arbitrary wave function must be as great as the lowest eigenvalue of the Hamiltonian.‡ To use this principle to determine the ground state of a system, we take a simple class of trial wave functions, and find the member of the class which gives the lowest expectation value of the Hamiltonian H . We can certainly get an upper bound to the ground state energy in this way. It is hoped that, if the trial wave functions are well chosen, the one which gives the lowest energy will have a useful resemblance to the ground state wave function.

  For an N -fermion system, the simplest trial wave function is a determinant

  where i and j run from 1 to N . A necessary condition for the expectation value of H to be a minimum is that it should be stationary, so that we have

  to first order in δ Φ*. A first-order change in Φ can be made by replacing one of the

  φi

  in Eq. (3.1) by

  φi

  +

  ηφk

  , where

  φk

  is a single-particle wave function orthogonal to all the

  φj

  with , and η is a small number. In the language introduced in the last chapter, the level i is occupied and the level k is unoccupied in Φ. The new determinant is still normalized to unity, to first order in η . The condition (3.2) is clearly equivalent to the condition that H should have no matrix elements between Φ and a determinant which differs from Φ in one row only.

  Using second quantization, we can write the change in the wave function as If the Hamiltonian contains just the kinetic energy and a two-body interaction, we can write it as

  where V

  jj ′, ll ′

  = V

  j ′ j , l ′l

  . The expression on the left of Eq. (3.2) can be evaluated by using the anticommutation relations (2.9) and remembering that creation operators with suffixes less than or equal to N and annihilation operators with suffixes greater than N give zero when

  Sign up to read

  Learn more about book
• 

  eBook - PDF

  Foundations of Classical Mechanics

  • P. C. Deshmukh(Author)
  • 2019(Publication Date)
  • Cambridge University Press

    (Publisher)

  Feynman went on to develop the path integral approach to the quantum theory based on the principle of variation. The path integral approach to quantum mechanics provides an alternative formulation of the quantum theory; it is equivalent to Heisenberg’s uncertainty principle, and the Schrödinger equation. It has the capacity to describe a mechanical system and to account for how it evolves with time. The variational principle can be adapted to provide a backward integration of classical mechanics as an approximation toward the development of quantum theory. Newtonian formulation is not suitable for this purpose. Foundations of Classical Mechanics 184 More need not be said to provide motivation introducing this topic even in a first college course on mechanics. We shall therefore jump into the subject right away. For parts this first section, we shall closely follow the Reference [6]. It nearly feels like a deep conspiracy of nature that the principle of causality and determinism, and the variational principle, produce results that are completely equivalent. The former makes no use of the ‘variation principle’, and the latter makes no use of ‘force’. The ‘variational principle’ is also known as the ‘principle of extremum action’. The general mathematical framework for the development and application of this technique is the ‘calculus of variation’. Its beginnings can be traced to the solution provided by Isaac Newton to the famous brachistochrone problem, which was posed by Johann (also known as Jean or John) Bernoulli in 1696.

  Sign up to read

  Learn more about book
• 

  eBook - PDF

  The variation method in quantum chemistry

  • Saul Epstein(Author)
  • 2012(Publication Date)
  • Academic Press

    (Publisher)

  Chapter I / General Theory of the Variation Method 1. Some Background Principles of least this and maximum that have for a long time fascinated scientists and philosophers alike. 1 Moreover, even principles of stationary this or that, though they may have somewhat less philosoph-ical appeal, nonetheless continue to interest scientists because on the one hand they usually provide a very compact way of stating the mathematical essentials of a theory, and on the other provide a useful avenue to the development of new theories. Thus it is not surprising that in his first paper on wave mechanics [3], Schrödinger presented his equation in the form of a variation principle, and only later [4] revealed some of the background which led to his writing down of the equation in the first place. An obvious reason for the philosophical fascination of minimal (or maximal) principles is that they seem to make physical laws less arbitrary, more rational, suggest a purpose, etc. Indeed, it is tempting to take a variation principle quite literally and then imagine that the variations are actually taking place; that at each moment or at each point, the system, whatever it is, is sampling all sorts of possible behavior, with the actual behavior then being selected on the basis that it will make the least change in something, that is, it will make some quantity stationary. Moreover, Feynman [5], following on some earlier work of Dirac, has shown that in a very real sense quantum mechanics can be derived from classical mechanics by omitting the final selection processes and assuming that all behavior, not just the classical one, is possible. In a not dissimilar vein, Ruedenberg and co-workers [6] have argued that one can get real insight into the nature of chemical binding by imagining that as a molecule forms, it actually does, so to speak, try one wave function, 1 See, for example, Yourgrau and Mandelstam [1] and Born [2]. 1

  Sign up to read

  Learn more about book
• 

  eBook - ePub

  Variational Principles in Dynamics and Quantum Theory

  • Wolfgang Yourgrau, Stanley Mandelstam(Authors)
  • 2012(Publication Date)
  • Dover Publications

    (Publisher)

  §10

  Relation between Variational Principles and the Older Form of Quantum Theory

  By common consent, quantum mechanics is the most revolutionary and comprehensive achievement in modern theoretical physics. However, the great changes, systematized into a consistent and coherent theory between 1925 and 1928, came about only gradually during the first quarter of the century. Throughout this period, hypothesis after hypothesis, each bringing us nearer to the imposing mathematico-physical edifice of today, had to be abandoned owing to logical inconsistencies and failure to produce agreement with observation and experiment. In this section, we shall endeavour to outline the part which the concept of action has occupied in these developments.

  The early stages of quantum theory are well-known and warrant but brief recapitulation here. In 1900, Planck postulated that light could not be radiated continuously, but only in whole multiples of hv, where v is the frequency and h a new universal constant known as “Planck’s constant,” its most recently computed value being 6·624.10-27 erg sec. The introduction of this hypothesis was rendered necessary in order to explain the spectral distribution of black body radiation; the black body was supposed to contain simple harmonic oscillators whose energy could only be integral multiples of hv.

  Sign up to read

  Learn more about book
• 

  eBook - PDF

  The Method of Weighted Residuals and Variational Principles, with Application in Fluid Mechanics, Heat and Mass Transfer

  • Courtney Finlayson(Author)
  • 1972(Publication Date)
  • Academic Press

    (Publisher)

  Some of the variational principles are motivated by Hamilton’s principle, which governs the equations of mechanics and dynamics of discrete particles. The variational principle is the following. Make stationary the variational integral I subject to variations in the generalized coordinates qr which vanish at t, and t, : 12 I = ( L(qr, 4;) dt, L = T - V. (8.15) J t l T is the kinetic energy and V is the potential energy, which depends only on the generalized coordinates, qr, not their derivatives, 4,’. The Euler equation is easily found from Eq. (7.10), (8.16) 256 8 VARIATIONAL PRINCIPLES IN FLUID MECHANICS The form of the Lagrangian-the kinetic energy minus the potential energy- is common to many of the variational principles described below. Finally, throughout this chapter the variational shorthand (7.14) is used. 8.2 Variational Principles for Perfect Fluids Many attempts have been made to obtain the momentum equations from a variational principle patterned after Hamilton’s principle, which is so power- ful and useful in particle mechanics. These attempts have not all been suc- cessful except in the case of perfect fluids. As Truesdell and Toupin (1960, p. 595) point out, “the lines of thought which have led to beautiful variational statements for systems of mass-points have been applied in continuum me- chanics also, but only rarely are the results beautiful or useful.” We consider first the steady, irrotational flow of an incompressible fluid, then admit compressible fluids, and finally treat unsteady flow of compressible fluids. Some of this treatment is summarized in the elegant treatise by Serrin (1959a). Consider the steady irrotational flow of an incompressible fluid. Assume the body force is a potential function, pi? = -V@, and rearrange the con- vective terms in (8.1) to get (8.17) The last relation gives a form of Bernoulli’s theorem, (8.18) P +u - u + @ + - = constant, P which determines the pressure once the velocity is found.

  Sign up to read

  Learn more about book
• 

  eBook - PDF

  Ergodic Theory – Finite and Infinite, Thermodynamic Formalism, Symbolic Dynamics and Distance Expanding Maps

  • Mario Roy, Sara Munday, Mariusz Urbański(Authors)
  • 2021(Publication Date)
  • De Gruyter

    (Publisher)

  12 The variational principle and equilibrium states In Section 12.1, we state and prove a fundamental result of thermodynamic formalism known as the variational principle. This deep result establishes a crucial relationship between topological dynamics and ergodic theory, by way of a formula linking topo- logical pressure and measure-theoretic entropy. The variational principle in its classi- cal form and full generality was proved in [75] and [10]. The proof we present follows that of Michal Misiurewicz [49], which is particularly elegant, short, and simple. In Section 12.2, we introduce the concept of equilibrium states, give sufficient con- ditions for their existence, such as the upper semicontinuity of the metric entropy function (which prevails under any expansive system). We single out a special class of equilibrium states, those corresponding to a potential identically equal to zero, and following tradition, call them measures of maximal entropy. We do not deal in this chapter with the issue of the uniqueness of equilibrium states. Nevertheless, we pro- vide an example of a topological dynamical system with positive and finite topological entropy which does not have any measure of maximal entropy. 12.1 The variational principle For any topological dynamical system T : X → X, subject to a potential φ : X → ℝ and equipped with a T -invariant measure μ, the quantity h μ (T ) + ∫ φ dμ is called the free energy of the system T with respect to μ under the potential φ. The variational principle states that the topological pressure of a system is the supremum of the free energy generated by that system. Recall that M(T ) is the set of all T -invariant Borel probability measures on X and that by definition any potential φ is continuous. Theorem 12.1.1 (Variational principle). Let T : X → X be a topological dynamical sys- tem and φ : X → ℝ a potential. Then P(T , φ) = sup{h μ (T ) + ∫ X φ dμ : μ ∈ M(T )}.

  Sign up to read

  Learn more about book