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Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields

Minimum-uncertainty states and pseudoclassical dynamics

V. Canivell, Pere Seglar

Published Feb 15, 1977• 8 citations
Publisher

Abstract

All the minimum-uncertainty states $|z(\ensuremath{\alpha})〉$ are built as the eigenstates of generalized annihilation operators $a(\ensuremath{\alpha})$. We obtain the most general Hamiltonian which verfies the following property: Given an initial minimum-uncertainty state $|z(\ensuremath{\alpha})〉$ at $t = 0$, the ${z}^{\ensuremath{'}}$ which corresponds to the minimum-uncertainty state that maximizes ${|〈{z}^{\ensuremath{'}}(\ensuremath{\alpha})|z(\ensuremath{\alpha}),t〉|}^{2}$ follows the classical trajectory in phase space. Finally, we comment on its relationship with the most general Hamiltonian which preserves minimality.

DOI
10.1103/physrevd.15.1050

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All the minimum-uncertainty states ∣z(\ensuremathα)〉|z(\ensuremath{\alpha})〉∣z(\ensuremathα)〉 are built as the eigenstates of generalized annihilation operators a(\ensuremathα)a(\ensuremath{\alpha})a(\ensuremathα). We obtain the most general Hamiltonian which verfies the following property: Given an initial minimum-uncertainty state ∣z(\ensuremathα)〉|z(\ensuremath{\alpha})〉∣z(\ensuremathα)〉 at t=0t = 0t=0, the z\ensuremath′{z}^{\ensuremath{'}}z\ensuremath′ which corresponds to the minimum-uncertainty state that maximizes ∣〈z\ensuremath′(\ensuremathα)∣z(\ensuremathα),t〉∣2{|〈{z}^{\ensuremath{'}}(\ensuremath{\alpha})|z(\ensuremath{\alpha}),t〉|}^{2}∣〈z\ensuremath′(\ensuremathα)∣z(\ensuremathα),t〉∣2 follows the classical trajectory in phase space. Finally, we comment on its relationship with the most general Hamiltonian which preserves minimality.
The set of initial conditions for which the pseudoclassical evolution algorithm (and minimality conservation) is verified for Hamiltonians of degrees NNN (N>2N>2N>2) is explicitly determined through a class of restrictions for the corresponding classical trajectories, and it is proved to be at most denumerable. Thus these algorithms are verified if and only if the system is quadratic except for a set of measure zero. The possibility of time-dependent $\ensuremath{\alpha}$-equivalence classes is studied and its physical interpretation is presented. The implied equivalence of the pseudoclassical and Ehrenfest algorithms and their relationship with minimality conservation is discussed in detail. Also, the explicit derivation of the general unitary operator which linearly transforms minimum-uncertainty states leads to the derivation, among others, of operators with a general geometrical interpretation in phase space, such as rotations (parity, Fourier).
We first review the properties of the harmonic-oscillator coherent states which can be equivalently defined as (a) a specific subset of the $x\ensuremath{-}p$ minimum-uncertainty states, (b) eigenstates of the annihilation operator, or (c) states created by a certain unitary exponential displacement operator. Then we present a new method for finding coherent states for particles in general potentials. Its basis is the desire to find those states which most nearly follow the classical motion, but it is most nearly a generalization of the minimum-uncertainty method. The properties of these states are discussed in detail. Next we show that the annihilation operator and displacement operator methods, as heretofore defined, cannot be applied to general potentials (whose eigenvalues are not equally spaced). We define a generalization of these methods but show that the states so defined are not, in general, equivalent to the minimum-uncertainty coherent states. We discuss a number of properties of our coherent states and the procedures we have used.
Both the coherent states and also the squeezed states of the harmonic oscillator have long been understood from the three classical points of view: the 1) displacement operator, 2) annihilation- (or ladder-) operator, and minimum-uncertainty methods. For general systems, there is the same understanding except for ladder-operator and displacement-operator squeezed states. After reviewing the known concepts, I propose a method for obtaining generalized minimum-uncertainty squeezed states, give examples, and relate it to known concepts. I comment on the remaining concept, that of general displacement-operator squeezed states.
Quantum dynamics of integrable systems is discussed. Localized wave packets generalizing the conventional coherent states of minimal uncertainty are constructed. The wave packet moves along a certain trajectory and does not change its shape for times of order 1 ¯h. In this Letter we suggest that the definition of a coherent state should be related to the Hamiltonian.The conventional coherent states [1] are defined as eigenstates of the annihilation operator α | z> = z | z> (1) The real and the imaginary parts of the complex variable z can be expressed by the mean values of the position ˆx and momentum ˆp operators in the coherent state. Under the oscillator Hamiltonian evolution Ut Ut|z> = |exp(−iωt)z> (2) where ω is the oscillator ′ √ s frequency. Then, because the state | z> is con-2¯h centrated around x = ω Rez it follows that Ut|z> has its support on the
We consider additional aspects of the recently derived “minimum uncertainty” (μ) wavelets. In particular, we show that they are fundamentally related to both the harmonic oscillator eigenstates and the canonical coherent states that play a fundamental role in quantum dynamics. In addition, we derive new raising and lowering operators that apply to the μ-wavelets. Finally, we explore in some detail the senses in which the μ-wavelets form complete sets that can be used in a variety of applications in quantum dynamics.