Poisson Bracket

6 Key excerpts on "Poisson Bracket"

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  Solid State and Quantum Theory for Optoelectronics

  • Michael A. Parker(Author)
  • 2009(Publication Date)
  • CRC Press

    (Publisher)

  4.4.1 D EFINITION OF THE P OISSON B RACKET AND R ELATION TO THE C OMMUTATOR We fi rst de fi ne the Poisson Brackets using the ‘‘ [ . . . ] ’’ similar to the commutator discussed in Chapter 3. However, the classical Poisson Brackets involve derivatives of functions where as the quantum mechanical commutators do not have this general form. De fi nition: Let A ¼ A ( q i , p i ) B ¼ B ( q i , p i ) be two differentiable functions of the generalized coordinates and momentum. We de fi ne the Poisson Brackets by [ A , B ] ¼ X i q A q q i q B q p i q B q q i q A q p i ! y 2 y 1 m 1 m 2 R θ FIGURE 4.7 The pulley system. Fundamentals of Classical Mechanics 213 Sometimes we subscript the brackets with q , p [ A , B ] ¼ [ A , B ] q , p The Poisson Bracket and commutator appear similar (when one ignores the fact that Poisson Brackets have derivatives) and provide somewhat similar formulations for the dynamics of a system. In the quantum theory, operators replace the classical dynamical variables (e.g., p ’ s and q ’ s). In fact, one starting method for fi nding the quantum Hamilton consists of determining the classical Hamiltonian, then substituting operators for the classical dynamical variables, and then specifying the commutators for those operators. Chapter 5 will show how the Heisenberg quantum picture is the closest analog to classical mechanics because the operators carry the system dynamics. In quantum theory, the commutation relations give time derivatives of operators where recall that the commutator is de fi ned by [  , ^ B ] ¼  ^ B ^ B with  , ^ B as operators. In the classical theories, the Hamiltonian uses functions for the dynamical variables (such as momentum p ) and the quantum theory replaces the functions with operators (such as ^ p ). Both the commutation relations and Poisson Brackets determine the evolution of the dynamical variables.

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  Lectures on Quantum Mechanics

  A Primer for Mathematicians

  • Philip L. Bowers(Author)
  • 2020(Publication Date)
  • Cambridge University Press

    (Publisher)

  We will consider this correspondence in the two lectures following. For two differentiable functions A and B of the generalized position and momentum coordinates q i and p i , the Poisson Bracket of A and B is {A, B} = N  i=1  ∂A ∂q i ∂B ∂p i − ∂A ∂p i ∂B ∂q i  . Quick and easy examples are {q i , p j } = δ ij and {q i , q j } = {p i , p j } = 0 for 1 ≤ i, j ≤ N . If A has an explicit dependence on time then, along a stationary tra- jectory q = q(t) of the system, A has the form A(q(t), p(t), t). Applying Hamilton’s equations, its total time derivative is easily seen to be given by the classical evolution equation dA dt = ∂A ∂t + {A, H}, (6.4) which expresses the time rate of change along the trajectory of a me- chanical quantity evaluated from the system. Applying this with A = q i and A = p i reproduces Hamilton’s equations aesthetically as ˙ q i = {q i , H} and ˙ p i = {p i , H} for i = 1, . . . , N. The evolution equation gives a useful condition for determining when a physical quantity is constant along a stationary solution. If A does not depend explicitly on time, so that ∂A/∂t = 0, then A is constant along a stationary trajectory q = q(t) precisely when the Poisson Bracket 6.4 Noether’s Theorem 91 {A, H} = 0. For example, as {H, H} = 0, the Hamiltonian is conserved in the system exactly when it does not depend explicitly on time. Easy calculations show that the Poisson Bracket satisfies {A, A} = 0, is antisymmetric in that {A, B} = −{B, A}, and linear in both A and B. A somewhat involved calculation, whose verification is left to the reader, shows that the Jacobi identity holds for any three C 2 functions A, B, and C , viz., that {A, {B, C }} + {B, {C, A}} + {C, {A, B}} = 0, demonstrating that the Poisson Bracket makes the vector space of real- valued C 2 functions of q and p into a real Lie algebra. A final comment on the Hamiltonian formalism is in order.

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  Essentials of Hamiltonian Dynamics

  • John H. Lowenstein(Author)
  • 2012(Publication Date)
  • Cambridge University Press

    (Publisher)

  2 The Hamiltonian formalism Having introduced the Hamiltonian formulation of classical dynamics, and applied it within a number of familiar contexts, we now turn to a systematic study of its salient features. Not least of these is the invariance of the dynamical equations with respect to the broad class of canonical transformations, whose defining property is the preservation of a certain antisymmetric, bilinear form, the Poisson Bracket. 2.1 The Poisson Bracket For a dynamical system with Hamiltonian H (q , p, t ), q = (q 1 , . . . , q n ), p = ( p 1 , . . . , p n ), the instantaneous state at time t , (q (t ), p(t )), evolves according to Hamilton’s equations (1.11) and (1.12). These also determine the evolution of any scalar function F (q , p, t ): d dt F (q (t ), p(t ), t ) = ∂ F ∂ q k ˙ q k + ∂ F ∂ p k ˙ p k + ∂ F ∂ t = [ F, H ] + ∂ F ∂ t , where the Poisson Bracket [ A, B ] of arbitrary A and B is defined as [ A, B ] = ∂ A ∂ q k ∂ B ∂ p k − ∂ A ∂ p k ∂ B ∂ q k . Here we have adopted the summation convention of summing over repeated indices from 1 to n. In terms of the 2n-dimensional vector ξ = (q 1 , . . . , q n , p 1 , . . . , p n ), we have [ A, B ] = ∇ ξ A ·  · ∇ ξ B, (2.1) 29 30 The Hamiltonian formalism where  is given by (1.13). Where more than one coordinate system is present, we will often write [ A, B ] q , p or [ A, B ] ξ in place of [ A, B ] to avoid ambiguity. The Poisson Bracket is a central concept of the Hamiltonian formalism which we shall work with in this book. We note that the Poisson Bracket is not only a linear function of each argument, but also antisymmetric under interchange of the two arguments, [ A, B ] = −[ B, A]. This ensures that the time evolution of the Hamiltonian itself is exceedingly simple, ˙ H = [ H, H ] + ∂ H ∂ t = ∂ H ∂ t , a result we obtained earlier.

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  Quantum Mechanics: A Complete Introduction: Teach Yourself

  • Alexandre Zagoskin(Author)
  • 2015(Publication Date)
  • Teach Yourself

    (Publisher)

  . Remember that when we take a partial derivative with respect to one independent variable, we keep all others constant. Then indeed

  Moreover, for any function f of x, px (e.g. a probability distribution function) its time derivative will be, according to the chain rule and the Hamilton’s equations, a Poisson Bracket of this function with the Hamilton function:

  If there are many pairs of conjugate variables, the formula remains the same. Only the Poisson Bracket will now be the sum of contributions from all these pairs.

  The important physical insight from this equation is that the Hamilton function – that is, energy – determines the time evolution of any physical system. Time and energy turn out to be related in classical mechanics, almost like (but not quite) the position and momentum.

  What has this to do with quantum mechanics? A while ago Werner Heisenberg found out that quantum equations of motion for operators (Heisenberg equations) have the following form:1

  In particular, . Here the brackets denote the commutator of two operators.

  Key idea: Rotations

  A vector r can be rotated to become vector r’. This operation can be written as the action of the rotation operator Ô on vector r:

  It transforms the old coordinates (e.g. (x,y,z)) into the new ones (x’,y’,z’).

  This is a particularly simple operation in a two-dimensional space; a planar rotation by an angle α. Then the vector r can be represented as a column of two numbers, x and y, and the operator Ô as a two-by-two matrix.

  The Hamiltonian operator Ĥ is the quantum operator, which corresponds to the classical Hamilton function, i.e. the energy of the system. For example, for a harmonic oscillator it is simply

  One of the great insights of Paul Dirac was that these Heisenberg equations of motion are directly analogous to classical equations of motion written using Poisson Brackets. One only needs to replace the classical functions and variables with quantum operators, and instead of a classical Poisson Bracket put in the commutator divided by iħ:2

  And voilà! You have a quantum equation of motion. This is a much more consistent scheme of quantization than the one of Bohr–Sommerfeld.

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  Free Theory

  • Anders Bengtsson(Author)
  • 2020(Publication Date)
  • De Gruyter

    (Publisher)

  If a classical mechanical theory is given in the Hamiltonian formulation, then the transition from classical Poisson Brackets {⋅, ⋅} to quantum commutators [⋅, ⋅] is done through the convention: If classically: { A , B } = C , then quantum mechanically: [ ̂ A , ̂ B ] = i ℏ ̂ C (3.73) where the Poisson Bracket is defined in formula (3.29). In terms of the phase space variables q n and p n obeying { q n , p m } = δ m n , we have [ ̂ q n , ̂ p m ] = i ℏ δ m n (3.74) In wave mechanics à la Schrödinger, the operators are realized as ̂ p n = − i ℏ 𝜕 𝜕 q n and ̂ q n = q n (3.75) The time evolution of a classical dynamical variable F turns into the time evolution of the corresponding quantum operator F according to ̇ F = { F , H } → i ℏ dF dτ = [ F , H ] (3.76) This corresponds to the Heisenberg picture where the time evolution of the quantum system is carried by the operators, and the states are constant in time. The Schrödinger equation, given above in formula (3.62), corresponds to the Schrödinger picture where the time evolution is carried by the states, and the operators are constant in time. 3.3.3 The Siegel mechanics to field theory algorithm We will very briefly review a simple instance of a method – based on BRST-symmetry – of passing from a mechanical model to a corresponding field theory. It was invented 132 | 3 Concepts, mathematical structures and notation by W. Siegel, and clarified by E. Witten, in connection with work on string field theory (see Section 2.11.1). The basic intuition behind the method may be argued to go back to L. de Broglie and E. Schrödinger. It is also implicit in Wigner’s work on wave equations discussed above (see in particular Section 2.3.2). The method is further developed in [241, 242] and reviewed in [187]. Indeed, referring back to the discussion at the beginning of Section 2.1, a classical free relativistic massless particle “moves” subject to the constraint p 2 = 0.

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  Quantum Social Science

  • Emmanuel Haven, Andrei Khrennikov(Authors)
  • 2013(Publication Date)
  • Cambridge University Press

    (Publisher)

  However, matrices in quantum theory are of infinite size. We shall explain later in the book why one cannot proceed with matrices of finite size. In such a case, some Hermitian matrices cannot be diagonalized. Besides eigenvalues, their spectra can contain a non-discrete part. It can even happen that there are no eigenvalues at all, and then such spectra are called continuous. This mathemat- ical formalism matches the physical situation: some physical observables, such as the particle’s position and momentum, are still continuous (as it is in classical mechanics). 1.13.1 Canonical commutation relations Heisenberg performed a formal translation of classical Hamiltonian mechanics (in which observables were given by functions on the phase space) into a new type of mechanics (quantum mechanics), in which observables are represented by Hermitian matrices. He correctly noted the crucial role the Poisson Brackets could play in classical formalism. They are defined for any pair of classical observables, f 1 , f 2 . We remark that Poisson Brackets are antisymmetric {f 1 , f 2 } = −{f 2 , f 1 }. A natural operation on the set of Hermitian matrices corresponding to Poisson Brackets is the commutator of two matrices: [A 1 , A 2 ] = A 1 A 2 − A 2 A 1 , (1.44) defined with the aid of standard matrix multiplication. Note that the commutator is anti-symmetric as well. The transformation of the usual multiplication of functions into matrix multiplication was the great contribution of Heisenberg towards the creation of quantum formalism. Starting with the classical position and momentum observables q j , p k (coordinates on the phase space) and the equalities for their Poisson Brackets, see (1.22), (1.23), he postulated that corresponding quantum observables denoted by ˆ q j , ˆ p k (hats are used to distinguish classical and quantum 10 Please see Chapter 4 where various linear algebra concepts are dealt with in more detail.

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