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Ehrenfest's Theorem and Nonclassical States of Light

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Ehrenfest's Theorem and Nonclassical States of Light GENERAL ARTICLE Ehrenfest’s Theorem and Nonclassical States of Light 1. Ehrenfest’s Theorem in Quantum Mechanics Lijo T George, C Sudheesh, S Lakshmibala and V Balakrishnan When it was ¯rst enunciated, Ehrenfest's The- orem provided a necessary and important link between classical mechanics and quantum me- chanics. Today, the content of the theorem is 1 2 understood to be a natural and immediate con- sequence of the equation of motion for opera- tors when quantum mechanics is formulated in 3 4 the Heisenberg picture. Nevertheless, the theo- rem leads to useful approximations when systems 1 Lijo T George got his with Hamiltonians of higher than quadratic or- Masters degree in physics from IIT Madras, and is der in the dynamical variables are considered. presently a doctoral student In this two-part article, we use it to provide a at RRI, Bangalore. His convenient illustration of the di®erences between research interests include so-called classical and nonclassical states of radi- quantum mechanics and ation. cosmology. 2C Sudheesh is with the 1. Paul Ehrenfest and the Quantum{Classical Department of Physics, Connection (IIST), Thiruvanan- thapuram. His research The Austrian-Dutch physicist Paul Ehrenfest (1880{1933) interests are nonlinear stands out among those who worked in the grey area dynamics, quantum optics of semiclassical physics straddling quantum mechanics and quantum information. (QM) and classical mechanics (CM). His work provided 3S Lakshmibala is with the a tangible correspondence between the quantum and Department of Physics, IIT Madras. Her research classical worlds, through the Ehrenfest theorem. This interests are dynamical theorem brought comfort to the practitioners of physics systems, quantum dynamics at a time in history when glimpses of the quantum world and quantum optics. and its laws left them uneasy, in marked contrast to 4 V Balakrishnan is with the the familiar terrain of classical physics. It connects the Department of Physics, dynamics of the expectation values of operators repre- IIT Madras. His research senting the physical observables of a quantum system to interests are statistical physics, stochastic the dynamics of their classical counterparts. The pur- processes and dynamical pose of this article is to illustrate this theorem in action systems. RESONANCE January 2012 23 GENERAL ARTICLE In CM, the state of a with the help of a simple model in quantum optics, and particle moving in to bring out precisely how nonclassical e®ects show up one dimension is in the case of certain experimentally realizable states of completely deter- quantized radiation. mined if the As several diverse concepts are involved in this task, this instantaneous values article is written in two parts. In this ¯rst part, we focus of its position and on the Ehrenfest theorem and its implications in QM. In momentum are the second part, we shall consider the instructive illus- known through direct tration of the theorem (in its general form) provided by or indirect measure- the dynamics of some states of a single-mode radiation ment. QM, on the ¯eld. other hand, is inherently probabi- We ¯rst set the stage with a brief recapitulation of some listic. We can only standard material in elementary QM. Even the sim- speak of the plest quantum mechanical system, a trailer to the quan- probability amplitude tum world, turns the strictly deterministic approach to for the position of the physics on its head. In CM, the state of a particle mov- ing in one dimension is completely determined if the particle to lie in the instantaneous values of its position and momentum are range (x, x + d x), known through direct or indirect measurement. QM, on when it is in a certain the other hand, is inherently probabilistic. We can only state at time t. speak of the probability amplitude for the position of the particle to lie in the range (x ; x + dx), when it is in a certain state at time t. The latter is speci¯ed by a state vector ª(t) in an appropriate Hilbert space. The probability ajmpliitude required is de¯ned as the overlap of the state vector with the position eigenstate x , and is given by x ª(t) . It is precisely this quantityj tihat is h j i called the position-space wave function Ã(x; t). Alter- natively, one may consider the probability amplitude for the particle to have a momentum in the range (p ; p+dp) when it is in the same state ª(t) . This is given by the overlap of the state vector wj ith ithe momentum eigen- state p , namely, the momentum-space wave function j i p ª(t) Ã(p; t). Both these wave functions are gener- Keywords h j i ´ Ehrenfest, expectation values, ally complex-valued functions of their arguments. They 2 2 quantum dynamics, quantum- yield the preobability densities Ã(x; t) and Ã(p; t) in j j j j classical correspondence. the position and momentum, respectively. The normal- e 24 RESONANCE January 2012 GENERAL ARTICLE ization In the Schrödinger picture of QM, 1 2 1 2 ª(t) ª(t) = Ã(x; t) dx = Ã(p; t) dp = 1 dynamics is h j i j j j j Z¡1 Z¡1 (1) incorporated through e ensures that the total probability (of the existence of timeevolution of the the particle) is unity, as required. In what follows, we state of the system as shall assume that the states we deal with are always given by the normalized to unity. Schrödinger equation. In the SchrÄodinger picture of QM, dynamics is incorpo- rated through time evolution of the state of the system as given by the SchrÄodinger equation d i~ ª(t) = H^ (t) ª(t) ; (2) dtj i j i where H^ is the Hamiltonian operator, assumed to be hermitian (more precisely, self-adjoint). An overhead caret will be used to denote operators. The adjoint of ª(t) , namely, the bra vector ª(t) , satis¯es the equa- tjion i h j d i~ ª(t) = ª(t) H^ (t): (3) ¡ dth j h j Note that we have allowed for a possible explicit time- dependence of the Hamiltonian. In the special case of an isolated system, H^ is t-independent. In that case the formal solution of the SchrÄodinger equation, for a given initial state ª(0) , is j i iH^ t= ª(t) = e¡ ~ ª(0) : (4) j i j i iH^ t= The operator e¡ ~ is unitary. This means that the state of the system evolves by the gradual unfolding of a unitary transformation of the initial state. The last The state of the statement remains true even when H^ has an explicit t- system evolves by dependence, although the unitary time-development op- erator is then more complicated than the simple expo- the gradual unfolding iH^ t= of a unitary nential e¡ ~. The unitarity of the time-development operator ensures the conservation of total probability, transformation of the i.e., ª(t) ª(t) = 1 at all times. initial state. h j i RESONANCE January 2012 25 GENERAL ARTICLE In contrast to a The values of physical quantities such as the position classical system, a and momentum of the particle in any given state are the measurement of outcomes of measurements of the corresponding quan- any observable  tum observables. The latter are operators that act on of a quantum the states of a system. The measurement problem in mechanical system QM is closely linked to the interpretation of QM and will, in general, other foundational questions. We do not enter into these change the state of matters here, but rather, restrict ourselves to the stan- the system. The dard, conventional view. In contrast to a classical sys- result of the tem, a measurement of any observable A^ of a quantum measurement can mechanical system will, in general, change the state of be any one of the the system. The result of the measurement can be any ^ eigenvalues of the one of the eigenvalues of the operator A. An amazing operator Â. feature of quantum physics is that the outcome of a mea- surement does not imply that the observable had that same value just prior to the measurement. In general, no de¯nitive statement can be made about the value of an observable before a measurement of that observ- able. What QM does yield, unambiguously, is complete information of a statistical nature about any physical observable. For instance, it provides us with an elegant formula for the average value, or expectation value A^ , of a quantum observable. It also gives us de¯nitehfori- mulas for the scatter about this average value, in terms ^ n QM provides us with of the expectation values of all the powers A , where an expectation value n = 2; 3; : : : . A^ , of a quantum h i We have stated that a single measurement of an observ- observable. It also able A^ yields any one of the eigenvalues of that oper- gives us definite ator. In order to obtain its expectation value A^ , the formulas for the same measurement has to be made, in principle,honi each scatter about this member of an in¯nite number of identical copies of the average value, in quantum system, all prepared in the same state. A^ h i terms of the is the arithmetic average of the results of all the mea- expectation values surements. As mentioned already, this quantity is cal- of all the powers culable in QM. For example, the expectation values of the position and momentum of a particle moving in one A^ n , where dimension, when it is in the normalized state ª(t) , are n = 2,,3,...
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