Hamiltonian

12 Key excerpts on "Hamiltonian"

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  eBook - ePub

  Quantum Mechanics

  Non-Relativistic Theory

  • L D Landau, E.M. Lifshitz(Authors)
  • 1981(Publication Date)
  • Butterworth-Heinemann

    (Publisher)

  CHAPTER II

  ENERGY AND MOMENTUM

  §8 The Hamiltonian operator

  The wave function Ψ completely determines the state of a physical system in quantum mechanics. This means that, if this function is given at some instant, not only are all the properties of the system at that instant described, but its behaviour at all subsequent instants is determined (only, of course, to the degree of completeness which is generally admissible in quantum mechanics). The mathematical expression of this fact is that the value of the derivative ∂Ψ/∂t of the wave function with respect to time at any given instant must be determined by the value of the function itself at that instant, and, by the principle of superposition, the relation between them must be linear. In the most general form we can write

  (8.1)

  where Ĥ is some linear operator; the factor i ħ is introduced here for a reason that will become apparent.

  Since the integral ∫Ψ*Ψ dq is a constant independent of time, we have

  Substituting here (8.1) and using in the first integral the definition of the transpose of an operator, we can write (omitting the common factor

  ii

  ħ)

  Since this equation must hold for an arbitrary function Ψ, it follows that we must have identically Ĥ = Ĥ; the operator Ĥ is therefore Hermitian. Let us find the physical quantity to which it corresponds. To do this, we use the limiting expression (6.1) for the wave function and write

  the slowly varying amplitude a need not be differentiated. Comparing this equation with the definition (8.1) , we see that, in the limiting case, the operator Ĥ reduces to simply multiplying by −∂S /∂t . This means that − ∂S /∂t is the physical quantity into which the Hermitian operator Ĥ passes.

  The derivative −∂S /∂t is just Hamilton’s function Ĥ for a mechanical system. Thus the operator Ĥ is what corresponds in quantum mechanics to Hamilton’s function; this operator is called the Hamiltonian operator or, more briefly, the Hamiltonian of the system. If the form of the Hamiltonian is known, equation (8.1) determines the wave functions of the physical system concerned. This fundamental equation of quantum mechanics is called the wave equation

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  Analytical Mechanics

  • Nivaldo A. Lemos(Author)
  • 2018(Publication Date)
  • Cambridge University Press

    (Publisher)

  As a consequence H = T + V = E , (7.11) that is, the Hamiltonian is the total energy expressed as a function of the coordinates and momenta. Conditions (1) and (2) prevail in the vast majority of physically interesting cases, so the Hamiltonian has the extremely important physical meaning of being the total energy in most situations of physical relevance. Furthermore, these conditions are only sufficient, and even if they are not satisfied it is possible that H coincide with the total energy (see Example 7.2 below). Example 7.1 Obtain Hamilton’s equations for a particle in a central potential. Solution In spherical coordinates the Lagrangian is L = T − V = m 2  ˙ r 2 + r 2 ˙ θ 2 + r 2 sin 2 θ ˙ φ 2  − V (r) , (7.12) whence p r = ∂ L ∂ ˙ r = m ˙ r , p θ = ∂ L ∂ ˙ θ = mr 2 ˙ θ , p φ = ∂ L ∂ ˙ φ = mr 2 sin 2 θ ˙ φ . (7.13) Solving these equations for the velocities we find ˙ r = p r m , ˙ θ = p θ mr 2 , ˙ φ = p φ mr 2 sin 2 θ , (7.14) so that H = ˙ r p r + ˙ θ p θ + ˙ φ p φ − L = 1 2m  p 2 r + p 2 θ r 2 + p 2 φ r 2 sin 2 θ  + V (r) . (7.15) This Hamiltonian equals the total energy and Hamilton’s equations are ˙ r = ∂ H ∂ p r = p r m , ˙ θ = ∂ H ∂ p θ = p θ mr 2 , ˙ φ = ∂ H ∂ p φ = p φ mr 2 sin 2 θ , (7.16) 219 Hamilton’s Canonical Equations ˙ p r = − ∂ H ∂ r = p 2 θ mr 3 + p 2 φ mr 3 sin 2 θ − dV dr , ˙ p θ = − ∂ H ∂θ = p 2 φ cot θ mr 2 sin 2 θ , ˙ p φ = − ∂ H ∂φ = 0 . (7.17) According to a general remark previously made, Eqs. (7.16) are the inverses of (7.13). With their use Eqs. (7.17) become identical to the Lagrange equations generated by the Lagrangian (7.12). Example 7.2 Construct the Hamiltonian and Hamilton’s equations for a charged particle in an external electromagnetic field. Solution In Cartesian coordinates we have L = m 2 ( ˙ x 2 + ˙ y 2 + ˙ z 2 ) − eφ(r, t) + e c v · A(r, t) , (7.18) whence p x = ∂ L ∂ ˙ x = m ˙ x + e c A x , p y = ∂ L ∂ ˙ y = m ˙ y + e c A y , p z = ∂ L ∂ ˙ z = m ˙ z + e c A z .

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  Emergence of the Quantum from the Classical

  Mathematical Aspects of Quantum Processes

  • Maurice de Gosson(Author)
  • 2017(Publication Date)
  • WSPC (EUROPE)

    (Publisher)

  Chapter 1

  Hamiltonian Mechanics

  Hamiltonian mechanics is perhaps the most powerful classical theory ever; it allows the prediction of the motion of celestial bodies, of aeroplanes, and of particles in fluids. While Hamilton’s equations of motion are easily derived, in the simplest cases, from Newton’s second law, Hamiltonian mechanics is more than just a fancy way of doing Newtonian mechanics. Hamiltonian mechanics could already be found in disguise in the work of Lagrange in celestial mechanics. Namely, Lagrange discovered that the equations expressing the perturbation of elliptical planetary motion due to interactions could be written down as a simple system of partial differential equations (known today as Hamilton’s equations, but Hamilton was only six years old when Lagrange made his discovery!). It is however undoubtedly Hamilton who realized, some twenty four years later, the theoretical importance of Lagrange’s discovery and exploited it fully. We mention in passing that the notation H for a Hamiltonian function was proposed by Lagrange to honor Christiaan Huygens, and not Hamilton!

  Like the movement of a symphony, a Hamiltonian flow involves a total ordering which implies the whole movement: Past, present, and future are actively present in any one movement. When we are listening to music we are actually directly perceiving an implicate order. This order is active because it is continuously flowing in emotional responses which are inseparable from the flow itself. Similarly, the solutions of Hamilton’s equations are uniquely determined for all bounded times and all locations close to the original one, exactly as in the symphony metaphor: if we observe during a tiny time interval the motion of a particle moving under the influence of a Hamiltonian flow, we see an unfoldment of the totality of the flow, which is uniquely determined by the past — and the future!

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  Compendium of Biophysics

  • Andrey B. Rubin(Author)
  • 2017(Publication Date)
  • Wiley-Scrivener

    (Publisher)

  r, t), this operator is

  (24.9)

  where Δ is the Laplace operator that in Cartesian coordinates is

  (24.10)

  Wave function ψ(r, t) will be the product of two functions, one of them depending only on time and the other only on the coordinates:

  Consequently, the complete wave function of the stationary state of a quantum system is as follows:

  (24.11)

  As seen, the complete wave function (Equation (24.11) ) of the stationary state of a quantum system is time-dependent, where value E corresponds to the energy of the given stationary state of the system.

  Wave functions, obtained from the solution of the stationary Schrodinger equation for a hydrogen atom, are of principal importance in quantum mechanics. The wave function of the ground state of a hydrogen atom most frequently used for estimate calculations is

  (24.12)

  Principle of Superposition. An important condition underlying many quantum effects is the principle of superposition of states. This principle is a consequence of the Schrodinger equation as a linear differential equation. According to the theory of differential equations, any sum of particular solutions of a linear differential equation is also its solution. This sum is called a superposition, or linear combination, of partial solutions.

  In quantum mechanics, the principle of superposition of states is one of the fundamental principles. If a system can be found in states described by wave functions ψ1 , ψ2 , ψ3

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  Notes on Hamiltonian Dynamical Systems

  • Antonio Giorgilli(Author)
  • 2022(Publication Date)
  • Cambridge University Press

    (Publisher)

  A major role in the development of the theory of classical dynamical sys- tems is played by the Hamiltonian formulation of the equations of dynamics. This chapter is intended to provide a basic knowledge of the Hamiltonian formalism, assuming that the Lagrangian formalism is known. A reader al- ready familiar with the canonical formalism may want to skip the present chapter. The canonical equations were first written by Giuseppe Luigi Lagrangia (best known by the French version of his name, Joseph Louis Lagrange) as the last improvement of his theory of secular motions of the planets [140]. The complete form, later developed in what we now call Hamiltonian formalism, is due to William Rowan Hamilton [106][107][108]. A short sketch concerning the anticipations of Hamilton’s work can be found in the treatise of Edmund Taylor Whittaker [209], §109. In view of the didactical purpose of the present notes, the exposition in this chapter follows the traditional lines. The chapter includes some basic tools: the algebra of Poisson brackets and the elementary integration meth- ods. Many examples are also included in order to illustrate how to write the Hamiltonian function for some models, often investigated using Newton’s or Lagrange’s equations. 1.1 Phase Space and Hamilton’s Equations The dynamical state of a system with n degrees of freedom is identified with a point on a 2n-dimensional differentiable manifold, denoted by F , endowed with canonically conjugated coordinates (q, p) ≡ (q 1 , . . . , q n , p 1 , . . . , p n ). The object of investigation is the evolution of the state of the system. The man- ifold F was named a phase space by Josiah Willard Gibbs [75]. The evolution of the system is determined by a real-valued Hamiltonian function H : F → R through the vector field defined by Hamilton’s equations (also called canonical equations), 2 Hamiltonian Formalism (1.1) ˙ q j = ∂H ∂p j , ˙ p j = - ∂H ∂q j , j = 1, .

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  An Introduction to the Mathematical Structure of Quantum Mechanics

  A Short Course for Mathematicians

  • F Strocchi(Author)
  • 2005(Publication Date)
  • WSPC

    (Publisher)

  Kinematics. The configuration or the state of a classical Hamiltonian sys- tem is (assumed to be) described by a set of canonical variables { q , p } , q = (q1, ..., qn), p = ( p i , ..., p n ) , briefly by a point P = { q , p } E r = phase space manifold. For simplicity, in the following, we will confine our discussion to the case in which r is compact. This is, e.g., the case in which the system is confined in a bounded region of space and the energy is bounded. The physical quantities or observables of the system, clearly include the q’s and p’s and their polynomials and therefore, without loss of generality, we can consider as observables their sup-norm closure, i.e. (real) continuous functions f ( q , p ) E CR(r), (for a further extension see the remark after eq. (1.2.3)). Every state P determines the values of the observables on that state and conversely, by the Stone-Weierstrass and Urysohn theorems, any P E r is uniquely determined by the values of all the observables on it (duality relation between states and observables). Dynamics. The relation between the measurement of an observable f at an initial time t o and at any subsequent time t is given by the time evolution of the canonical variables The time evolution of the canonical variables is given by the Hamilton equations . dH . dH q = ---, p = --dP % ’ (1.2.2) 1.2 Classical Hamiltonian systems 11 where H = H ( q , p ) is the Hamiltonian. Under general conditions (typically if gradH is Lipschitz continuous), for any initial data, the above system of equations has a unique solution local in time, which can be extended to all times under general conditions, e.g. if the surfaces of constant energy are compact 3. The mathematical structures involved are the theory of functions (on phase space manifolds) and the theory of first order differential equations, defined by Hamiltonian flows on phase space manifolds.

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  Physics of Electronic Materials

  Principles and Applications

  • Jørgen Rammer(Author)
  • 2017(Publication Date)
  • Cambridge University Press

    (Publisher)

  Foremost, we note that, in quantum mechanics, probability has entered in a fundamental way, i.e. chance is a feature of how the world works. In general, for given identical circumstances, it is impossible to predict what will happen in the future: quantum mechanics is probabilistic in nature. Quantum mechanics only provides the odds for different outcomes. We also observe the strange feature that, in contrast to any physi-cal statement, a description in terms of complex numbers is demanded. For the interested reader, it is shown in Appendix A that the Schrödinger equation can be arrived at from a few basic principles. 1.1 Hamiltonian Consider the Schrödinger equation for a single particle of mass m in a potential, i ∂ψ ( x , t ) ∂ t = − 2 2 m ∂ 2 ∂ x 2 + V ( x , t ) ψ ( x , t ). (1.7) The Hamiltonian, specifying the Schrödinger equation, then consists of the Laplacian and a multiplication operator, the space-and time-dependent potential, V ( x , t ), multiplying the wave function, ˆ H ≡ ˆ H ( t ) = − 2 2 m ∂ 2 ∂ x 2 + V ( x , t ). (1.8) The real scalar potential, V ( x , t ), describes the fact that the particle is not free, but at dif-ferent locations experiences different environments, which in addition can be changing 1 In the same vein, you have probably solved Newton’s equation having the expression for the gravitational force being handed to you. 3 1.2 Free Propagator in time. As shown in Appendix A , the potential is the potential energy the particle has according to classical mechanics, and the gradient of the potential equals the classical force, F ( x , t ) = −∇ V ( x , t ). That normalization at one instant of time, d x | ψ ( x , t ) | 2 = 1, (1.9) guarantees it at all times is a defining property of a Hamiltonian. If a function is normal-ized, it vanishes spatially at infinity in order for the normalization integral to be finite.

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  Quantum Chemistry Simulation of Biological Molecules

  • Eudenilson L. Albuquerque, Umberto L. Fulco, Ewerton W. S. Caetano, Valder N. Freire(Authors)
  • 2021(Publication Date)
  • Cambridge University Press

    (Publisher)

  The time evolution of the state is found by solving Hamilton equations: dx dt = ∂H ∂p , (1.2) 1 2 Basic Properties of Quantum Chemistry − dp dt = ∂H ∂x , (1.3) or, considering the state vector S, d S dt = {S,H } , (1.4) where {A,B } is the Poisson bracket, {A,B } = ∂ A ∂x ∂B ∂p − ∂B ∂x ∂ A ∂p . (1.5) Thus, the Hamiltonian rules the evolution of the classical state S in time. As a matter of fact, this method can be extended to systems much more complex than the spring–mass system. If the system has N particles, with inde- pendent spatial coordinates (q 1 ,q 2 , . . . ,q N ) and respective conjugated momenta (p 1 ,p 2 , . . . ,p N ), the Hamiltonian is a function of the state: S = (q 1 ,q 2 , . . . ,q N ,p 1 ,p 2 , . . . ,p N ), (1.6) which evolves in time according to the Eq. (1.4) provided we redefine the Poisson bracket as {A,B } = N  i =1  ∂ A ∂q i ∂B ∂p i − ∂B ∂q i ∂ A ∂p i  . (1.7) At every moment, the classical system is completely described by the classical state S, and one can always in principle solve Hamilton equations to evaluate its past and future states. Classical physics is completely deterministic: knowing S at some time t 0 and the form of its Hamiltonian function, it is possible to disclose the whereabouts of the system at any given time t . This picture, however, is radically changed in quantum mechanics. In a quantum system, the physical state is described not by a position-momentum vector S but by a unit complex vector |  in an abstract geometric space called Hilbert space. The x and p coordinates of our spring-mass oscillator must be replaced by Hermitian operators ˆ x and ˆ p, whose eigenstates correspond, respec- tively, to states with well-defined values of the position x and the momentum p.

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  Principles of Quantum Electronics

  • Dietrich Marcuse(Author)
  • 2012(Publication Date)
  • Academic Press

    (Publisher)

  C H A P T E R R E V I E W O F Q U A N T U M M E C H A N I C S 1.1 H a m i l t o n i a n M e c h a n i c s Formulation of Hamiltonian Mechanics The transition from classical mechanics to quantum mechanics can be made most conveniently and easily by using the Hamiltonian formulation of classical mechanics. Before giving a brief outline of quantum mechanics we, therefore, review a few of the most important equations of the classical Hamiltonian mechanics. The methods of classical mechanics and quantum mechanics are vastly different. Classical mechanics is based on the assumption that any physically interesting variable connected with a particle, such as its position, its velocity, or its energy, can be measured with arbitrary precision and without mutual interference from any other such measurement. Classical mechanics, there-fore, uses sets of variables and functions of these variables to enable us to predict the behavior of physical systems by providing us with differential equations that determine the changes of these functions in space and time. Quantum mechanics is based on the realization that the measuring process may affect the physical system. It is, therefore, impossible in principle to measure simultaneously certain pairs of variables with arbitrary precision. The measurement of one variable affects other variables in such a way that it prevents us from knowing what their values might have been. The mathe-matical formulation of the laws of physics that takes this basic idea into account is very different from the mathematical formulation of classical mechanics, as we shall see later in this chapter. The laws of classical mechanics can be expressed in various mathematical forms. The simplest formulation is based upon Newton's law stating that the 1 2 Review of Quantum Mechanics CHAPTER ONE force acting on a body is equal to the product of its mass times its acceleration.

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  Quantum Mechanics for Hamiltonians Defined as Quadratic Forms

  • Barry Simon(Author)
  • 2015(Publication Date)
  • Princeton University Press

    (Publisher)

  CHAPTER II THE Hamiltonian II. 1. Introduction From a fundamental point of view, the basic dynamical object in a quantum mechanical system is the propagator, U(t), which is a one param- eter group of unitary operators. 1 On the other hand, the objects one ob- tains from classical mechanics are the Hamiltonian and its dynamics as determined by the Schrodinger equation. Stone's theorem (Theorem A.21) 2 provides the connection between the two approaches; on its authority we know that there is a one-to-one correspondence between unitary one- parameter groups and self-adjoint operators via: U(t) = e~ iHt The connection with Schrodinger's equation is provided by: ijj- (U(t)W) = HU(t)¥ for Ψ<τϋ(Η). Since Stone's theorem deals with self-adjointness rather than mere Hermiticity, a mathematically non-trivial question immediately arises: to Actually, the fundamental object is a group of automorphisms of the rays. How- ever, an analysis of Wigner and Bargmann [5, 6, 123, T28] allows one to show that the automorphisms have a realization as a continuous one parameter group of unitary operators. 2 Assorted mathematical theorems and definitions are collected in an appendix at the end of the monograph. 32 II. THE Hamiltonian show that in some sense the Hamiltonian is self-adjoint. One of the simplest and most elegant results along these lines is: THEOREM II. 1 (Kato [65]). Let Then and the operator with domain is self-adjoint. I This theorem depends on the Kato-Rellich theorem (Theorem A.7) and the simple fact that is Kato tiny relative (Definition A. 14) to This last observation has a proof analogous tc our discussion of Theorem 1.21: LEMMA II.2. If then Proof. 5 has a kernel where Since this kernel is Hilbert-Schmidt (Appendix, part (d)) with Schmidt norm . Thus, for any a, we can find E with In this chapter, we discuss a self-adjoint Hamiltonian, when By Corollary 1.2, this class includes the class covered by Kato.

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  Geometry, Topology and Physics

  • Mikio Nakahara(Author)
  • 2018(Publication Date)
  • CRC Press

    (Publisher)

  1 QUANTUM PHYSICS A brief introduction to path integral quantization is presented in this chapter. Physics students who are familiar with this subject and mathematics students who are not interested in physics may skip this chapter and proceed directly to the next chapter. Our presentation is sketchy and a more detailed account of this subject is found in Bailin and Love (1996), Cheng and Li (1984), Huang (1982), Das (1993), Kleinert (1990), Ramond (1989), Ryder (1986) and Swanson (1992). We closely follow Alvarez (1995), Bertlmann (1996), Das (1993), Nakahara (1998), Rabin (1995), Sakita (1985) and Swanson (1992). 1.1 Analytical mechanics We introduce some elementary principles of Lagrangian and Hamiltonian formalisms that are necessary to understand quantum mechanics. 1.1.1 Newtonian mechanics Let us consider the motion of a particle m in three-dimensional space and let x(t) denote the position of m at time t.x Suppose this particle is moving under an external force F(x). Then jc(0 satisfies the second-order differential equation called Newton’s equation or the equation of motion. If force F(x) is expressed in terms of a scalar function V(x) as F(x) = — VV(jr), the force is called a conserved force and the function V(x) is called the potential energy or simply the potential. When F is a conserved force, the combination ( 1 . 1 ) ( 1 . 2 ) is conserved. In fact, d E dxjc d2Xk SV dxk — / YYl -----------------------| ----------------------dt , ' dt d t 2 dxk dt l:=x,y,z L dxk dt dxjc I dt av d 5 = 0 1 We call a particle with mass m simply ‘a particle ra 1 2 QUANTUM PHYSICS where use has been made of the equation of motion. The function E , which is often the sum of the kinetic energy and the potential energy, is called the energy. Example 1.1. (One-dimensional harmonic oscillator) Let x be the coordinate and suppose the force acting on m is F (i) = —k x , k being a constant. This force is conservative. In fact, V(x) = k x 2 yields F ( x ) = —d V ( x ) / d x = —kx.

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  Introduction to Mathematical Physics

  • Michael T. Vaughn(Author)
  • 2008(Publication Date)
  • Wiley-VCH

    (Publisher)

  In terms of ( X, P ) , the Hamiltonian is H = 1 2 ω ( P 2 + X 2 ) (3.272) 148 3 Geometry in Physics We can further introduce variables J, α by X ≡ √ 2 J sin α P ≡ √ 2 J cos α (3.273) corresponding to J = 1 2 ( P 2 + X 2 ) tanα = P X (3.274) In terms of these variables, the Hamiltonian is given simply by H = ωJ (3.275) The variables J, α are action-angle variables. Since the Hamiltonian (3.275) is indepen-dent of the angle variable α , the conjugate momentum J (the action variable ) is a constant of the motion, and ˙ α = ∂H ∂J = ω (3.276) Thus the motion in the phase space defined by the variables ( X, P ) is a circle of radius √ 2 J , with angular velocity given by ˙ α = ω . Note that for the special case of simple harmonic oscillator, the angular velocity ˙ α is independent of the action variable. This is not true in general (see Problem 20). Exercise 3.22. Show that the transformation to action-angle variables is canonical, i.e., show that dJ ∧ dα = dP ∧ dX Then explain the choice of √ 2 J , rather than some arbitrary function of J , as the “radius” variable in Eq. (3.273). 3.6 Fluid Mechanics A real fluid consists of a large number of atoms or molecules whose interactions are suffi-ciently strong that the motion of the fluid on a macroscopic scale appears to be smooth flow superimposed on the thermal motion of the individual atoms or molecules, the thermal motion being generally unobservable except through the Brownian motion of particles introduced into the fluid. An ideal fluid is characterized by a mass density ρ = ρ ( x, t ) and a velocity field u = u ( x, t ) , as well as thermodynamic variables such as pressure p = p ( x, t ) and temperature T = T ( x, t ) . If the fluid is a gas, then it is often important to consider the equation of state relating ρ , p , and T . For a liquid, on the other hand, it is usually a good approximation to treat the density as constant ( incompressible flow ).

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