Ehrenfest Theorem

6 Key excerpts on "Ehrenfest Theorem"

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  Quantum Mechanics

  A Modern Development

  • Leslie E Ballentine(Author)
  • 1998(Publication Date)
  • WSPC

    (Publisher)

  14-1 Ehrenfest's Theorem and Beyond 391 a harmonic oscillator or a free particle. But if the width of the position proba-bility distribution is small compared to the typical length scale over which the force varies, then the centroid of the quantum-mechanical probability distri-bution will follow a classical trajectory. This is Ehrenfest's theorem. It is sometimes asserted that the conditions for classical behavior of a quan-tum system are just those required for Ehrenfest's theorem. But, in fact, Ehrenfest's theorem is neither necessary nor sufficient to define the classical regime (Ballentine, Yang, and Zibin, 1994). Lack of sufficiency — that a sys-tem 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 thermo-dynamic 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, 6Q = Q-(Q), (14.7) 6P = P- (P) , (14.8) 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), ^ f -!«(*>+!««»>& «•> + -. (.4.9) where the average position and momentum are qo = (Q) and po = (-P). If ((6Q) 2 ) and higher order terms are negligible, we recover Ehrenfest's theorem, with qo and po 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 ft.

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  Lectures on Quantum Mechanics

  • Ashok Das(Author)
  • 2012(Publication Date)
  • WSPC

    (Publisher)

  In other words, Ehrenfest’s theorem says that expectation values of operators in quantum states obey classical equations. ◮ Example. As a direct application of the above discussions to particle motion in one dimension, where H = P 2 2 m + V ( X ) , (3.58) 3.5 Ehrenfest Theorem 77 we can calculate the time evolution of the expectation value of the coordinate operator using (3.57), d angbracketleft X angbracketright d t = d d t angbracketleft ψ | X | ψ angbracketright = 1 i planckover2pi1 angbracketleft ψ | [ X,H ] | ψ angbracketright = 1 i planckover2pi1 angbracketleft ψ | bracketleftbigg X, P 2 2 m + V ( X ) bracketrightbigg | ψ angbracketright = 1 i planckover2pi1 angbracketleft ψ | bracketleftbigg X, P 2 2 m bracketrightbigg | ψ angbracketright = 1 i planckover2pi1 × 1 2 m angbracketleft ψ | P [ X,P ] + [ X,P ] P | ψ angbracketright = 1 i planckover2pi1 × 1 2 m × 2 i planckover2pi1 angbracketleft ψ | P | ψ angbracketright = 1 m angbracketleft ψ | P | ψ angbracketright = angbracketleft P angbracketright m . (3.59) Let us recall from (1.43) that in classical mechanics we have ˙ x = ∂H ∂p = p m , which can be compared with (3.59). Furthermore, the time evolution of the expectation value of the momentum operator is obtained to be d angbracketleft P angbracketright d t = d d t angbracketleft ψ | P | ψ angbracketright = 1 i planckover2pi1 angbracketleft ψ | [ P,H ] | ψ angbracketright = 1 i planckover2pi1 angbracketleft ψ | bracketleftbigg P, P 2 2 m + V ( X ) bracketrightbigg | ψ angbracketright = 1 i planckover2pi1 angbracketleft ψ | [ P,V ( X )] | ψ angbracketright = 1 i planckover2pi1 angbracketleft ψ | ( − i planckover2pi1 ) d V ( X ) d X | ψ angbracketright = −angbracketleft ψ | d V ( X ) d X | ψ angbracketright = −angbracketleft d V ( X ) d X angbracketright . (3.60) Once again, we note from (1.43) that in classical physics, ˙ p = − ∂H ∂x = − d V ( x ) d x , (3.61) which can be compared with (3.60).

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  Quantum Mechanics I

  A Problem Text

  • David DeBruyne, Larry Sorensen(Authors)
  • 2018(Publication Date)
  • Sciendo

    (Publisher)

  Chapter 6 Ehrenfest, Heisenberg, and Gauss Part 1, Ehrenfest’s Theorem Quantum mechanics must give the same results as classical mechanics in a classical regime. Bohr’s correspondence principle, which states the relations of quantum mechanics reduce to the relations of classical mechanics for large quantum numbers, is often cited. Paul Ehrenfest originated a different answer. Ehrenfest said replace the dynamical variables of classical mechanics with the expectation values of quantum mechanics and you obtain the same relations. 6–1. Derive Ehrenfest’s Theorem. This derivation assumes that an operator representing an observable quantity is time independent, and that the state vector is time dependent. Observable quantities are functions of position only in this text, and time evolution of a state vector has been discussed in previous chapters. Notice that the derivation is for a general operator A , which may represent position, momentum, energy, or any other observable quantity. Remember that a commutator is defined bracketleftbig A , B bracketrightbig = A B -B A . Start with the expectation value of a time independent operator, < A > = <ψ | A | ψ> , and take a time derivative. The wave function is assumed to be a function of time, so using the chain rule to take the derivative, d dt < A > = < ˙ ψ | A | ψ> + <ψ | ˙ A | ψ> + <ψ | A | ˙ ψ>. Since the operator is assumed to be time independent, the middle term is zero so this reduces to d dt < A > = < ˙ ψ | A | ψ> + <ψ | A | ˙ ψ> . (1) The Schrodinger postulate is H | ψ> = i ¯ h | ˙ ψ> ⇒ | ˙ ψ> = 1 i ¯ h H | ψ> = -i ¯ h H | ψ>. Forming the adjoint of the last relation, < ˙ ψ | = <ψ | H † parenleftbigg i ¯ h parenrightbigg , and since the Hamiltonian is Hermitian, this is < ˙ ψ | = i ¯ h <ψ | H .

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  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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  Introduction to Quantum Mechanics

  • Henrik Smith(Author)
  • 1991(Publication Date)
  • WSPC

    (Publisher)

  Under such circumstances one writes the Hamiltonian in a symmetric form in the classical variables before replacing these by operators, corresponding to the substitution ab —► (ab + 6d)/2. The general form of the time-dependent Schrodinger equation is thus Hil> = ih?£, (4.5) where the Hamiltonian H of the system under consideration is derived from 108 Introduction to Quantum Mechanics the classical Hamiltonian as described above, with the possible addition of relativistic correction terms. The relativistic equation for a single electron, known as the Dirac equation, is a linear equation like the Schrodinger equation, but the wave function has in this case four components, which means that (4.5) is replaced by a 4 x 4 matrix differential equation, cf. (1.140). In the limit where the relativistic effects may be treated as small correction terms, it is sufficient to take into account two of the four components of the wave function. This introduces the column vectors which will be used in Chapter 7 to represent the electron spin, cf. (2.79-80). 4.1 Ehrenfest's theorem Quantum mechanics reduces to classical mechanics in an appropriate limit. To elucidate the nature of this limit we shall derive an expression for the time derivative of an arbitrary physical quantity. It follows from the Schrodinger equation HtP = ih^ (4.6) by complex conjugation that HP = ~ ih ^f-(4-7) The time derivative of the expectation value < A >= (ipAtp) of an arbitrary physical quantity A is Here we have taken into account that the operator A may depend explicitly on time. By inserting (4.6-7) in (4.8) and using that H is Hermitian we obtain the result d < A > 1 . 2 r y. dA , A ^ -^ - = ^ < [ ^ H ] > + < -> . (4.9) This is EhrenfesVs theorem. We shall use (4.9) to demonstrate how the mean values of the momentum and the position of a particle in a suitable limit obey the classical equations of motion.

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  Advances in Quantum Mechanics

  • Paul Bracken(Author)
  • 2013(Publication Date)
  • IntechOpen

    (Publisher)

  Author details Yasuteru Shigeta 1,2 1 Department of Materials Engineering Science, Graduate School of Engineering Sciences, Osaka University, Machikaneyama-cho, Toyonaka, Osaka, Japan 2 Japan Science and Technology Agency, Kawaguchi Center Building, Honcho, Kawaguchi-shi, Saitama, Japan Quantal Cumulant Mechanics as Extended Ehrenfest Theorem http://dx.doi.org/10.5772/53703 313 References [1] Schrödinger, E. (1926). Quantisierung als Eigenwertproblem (Erste Mitteilung). An‐ nalen der Physik, Vol. 79, No. 4, (April, 1926), pp. 361-376. ISSN: 0003-3804 [2] Schrödinger, E. (1926). Quantisierung als Eigenwertproblem (Zweite Mitteilung). An‐ nalen der Physik, Vol. 79, No. 6, (May, 1926), pp. 489-527. ISSN: 0003-3804 [3] Schrödinger, E. (1926). Quantisierung als Eigenwertproblem (Dritte Mitteilung: Stor‐ ungstheorie, mit Anwendung auf den Starkeffekt der Balmerlinien). Annalen der Physik, Vol. 80, No. 13, (September, 1926), pp. 437-490. ISSN: 0003-3804. [4] Schrödinger, E. (1927). Quantisierung als Eigenwertproblem (Vierte Mitteilung). An‐ nalen der Physik, Vol 81, No. 18, (September, 1927), pp. 109-139. 0003-3804. [5] Heisenberg, W. (1943). The observable quantities in the theory of elementary parti‐ cles. III. Zeitschrift für physik, Vol 123, No. 1-2, (March, 1943) pp. 93-112. ISSN: 0044-3328. [6] Ehrenfest, P. (1927). Bemerkung über die angenäherte Gültigkeit der klassischen Mechanik innerhalb der Quantenmechanik. Zeitschrift für physik, Vol 45, No. 7-8, (July, 1927), pp. 455-472. ISSN: 0044-3328. [7] Prezhdo, O.V. & Pereverzev, Y.V. (2000). Quantized Hamilton dynamics. Journal of Chemical Physics, Vol. 113, No. 16, (October 22, 2000), pp. 6557-6565. ISSN: 0021-9606. [8] Prezhdo, O.V. (2006). Quantized Hamilton Dynamics. Theoretical Chemistry Ac‐ counts, Vol. 116, No. 1-3, (August 2006), pp. 206-218. ISSN: 1432-881X and references cited therein.

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