Ehrenfest Theorem

3 Key excerpts on "Ehrenfest Theorem"

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  eBook - ePub

  Quantum Mechanics

  A Modern Development

  • Leslie E Ballentine(Author)
  • 2014(Publication Date)
  • WSPC

    (Publisher)

  neither necessary nor sufficient to define the classical regime (Ballentine, Yang, and Zibin, 1994). Lack of sufficiency — that a system may obey Ehrenfest’s theorem but not behave classically — is proved by the example of the harmonic oscillator. It satisfies (14.6) exactly for all states. Yet a quantum oscillator has discrete energy levels, which make its thermodynamic properties quite different from those of the classical oscillator. Lack of necessity — that a system may behave classically even when Ehrenfest’s theorem does not apply — will be demonstrated below.

  Corrections to Ehrenfest’s theorem

  Let us introduce operators for the deviations from the mean values of position and momentum, and expand (14.1) and (14.2) in powers of these deviation operators. Taking the average in some chosen state then recovers (14.3), and yields, in place of (14.4),

  where the average position and momentum are q 0 = 〈Q 〉 and p 0 = 〈P 〉. If 〈(δQ )2 〉 and higher order terms are negligible, we recover Ehrenfest’s theorem, with q 0 and p 0 obeying the classical equations.

  The terms in (14.9) beyond F (q 0 ) are corrections to Ehrenfest’s theorem. But they are not essentially quantum-mechanical in origin, as is evidenced by the fact that they do not depend explicitly on ħ . Indeed, 〈(δQ )2 〉 is just a measure of the width of the position probability distribution, which need not vanish in the classical limit. The proper interpretation of these correction terms can be found by comparison with a suitable classical ensemble.

  Let

  ρc

  (q , p , t ) be the probability distribution in phase space for a classical ensemble. It satisfies the Liouville equation, which describes the flow of probability in phase space,

  From it, we can calculate the classical averages,

  Differentiating these expressions with respect to t , using (14.10), and integrating by parts as needed, we obtain

  which are the classical analogs of (14.3) and (14.4). Expanding (14.14) in powers of δq =

  q − qc

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  eBook - ePub

  Introductory Matter Physics

  • Francesco Simoni(Author)
  • 2018(Publication Date)
  • WSPC

    (Publisher)

  principle of correspondence. It also guarantees that the same conservation laws can be applied to quantum-mechanical systems. It means that the results of analysis coming from the statistical approach of quantum mechanics correspond to those of the deterministic approach of classical mechanics when measurement is based on a high number of single events.

  The mathematical demonstration of this principle is given by Ehrenfest’s theorem. We will first give the demonstration of Eq. (2.75), showing that for each component of the velocity the expectation value is the time derivative of the corresponding position coordinate, and get

  Let us do the calculation for the x component,

  taking into account that inside the integral x appears as a time-independent operator. From the Schrödinger equation,

  Then

  The ∇2 operator is self-adjoint:

  Then On the other hand, Therefore, which coincides with the first of Eqs. (2.93)

  Therefore, we can apply to expectation values the usual definition of velocity used in classical mechanics:

  Another fundamental equation is the second law of classical dynamics, F = ma, or

  which we can demonstrate to be fulfilled by the expectation values of the corresponding physical quantities. This can be done by using Eq. (2.91) to calculate the time derivative of the linear momentum p.

  According to Eq. (2.91), we write

  Since px is not explicitly dependent on t,

  Then On the other hand, since the derivation order is not relevant. As a consequence, i.e. This calculation can be repeated for the other components to give

  Therefore, we get

  This is a very important result of the theory, since it gives the link between the microscopic statistical approach of quantum mechanics and the macroscopic deterministic approach of Newtonian mechanics.

  2.6. Eigenfunctions and eigenvalues

  Looking at the Schrödinger equation independent of time (2.57),

  on the left we have the Hamiltonian operator applied to the wave function ϕ(r) and on the right a number E which corresponds to the total energy of the system described by ϕ(r) multiplied by the same wave function. In other words, we can say that the result of applying to ϕ(r) is the value of energy times ϕ(r). In such a case, we call all the functions ϕ(r) fulfilling this equation eigenfunctions of and we call eigenvalues the corresponding values of E

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  eBook - ePub

  Quantum Mechanics

  • Richard Fitzpatrick(Author)
  • 2015(Publication Date)
  • WSPC

    (Publisher)

  υ . Heisenberg’s equation of motion yields

  Thus, any observable that commutes with the Hamiltonian is a constant of the motion (hence, it is represented by the same fixed operator in both the Schrödinger and Heisenberg pictures). Only those observables that do not commute with the Hamiltonian evolve in time in the Heisenberg picture.

  3.4Ehrenfest Theorem

  We have now introduced all of the basic elements of quantum mechanics. The only element that is lacking is some rule to determine the form of the quantum mechanical Hamiltonian. For a physical system that possess a classical analog, we generally assume that the Hamiltonian has the same form as in classical physics (i.e., we replace the classical coordinates and conjugate momenta by the corresponding quantum mechanical operators). This scheme guarantees that quantum mechanics yields the correct classical equations of motion in the classical limit. Whenever an ambiguity arises because of non-commuting observables, this can usually be resolved by requiring the Hamiltonian H to be an Hermitian operator. For instance, we would write the quantum mechanical analog of the classical product

  x px

  , appearing in the Hamiltonian, as the Hermitian product (1/2)(

  x px

  +

  px x

  ). When the system in question has no classical analog then we are reduced to guessing a form for H that reproduces the observed behavior of the system.

  Consider a three-dimensional system characterized by three independent Cartesian position coordinates

  xi

  (where i runs from 1 to 3), with three corresponding conjugate momenta

  pi

  . These are represented by three commuting position operators

  xi

  , and three commuting momentum operators

  pi

  , respectively. The commutation relations satisfied by the position and momentum operators are

  [See Equation (2.25).] It is helpful to denote (x 1 , x 2 , x 3 ) as x and (p 1 , p 2 , p 3 ) as p

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