arXiv subjects
Publications and source records attributed to Michael Martin Nieto.
Using the Paul Trap as a model, we point out that the same wave functions can be variously coherent or squeezed states, depending upon the system they are applied to.
We describe a simple volcano potential, which is supersymmetric and has an analytic, zero-energy, ground state. (The KK modes are also analytic.) It is an interior harmonic oscillator potential properly matched to an exterior angular momentum-like tail. Special cases are given to elucidate the physics, which may be intuitively useful in studies of higher-dimensional gravity.
In this paper, we attack the specific time-dependent Hamiltonian problem H=-{1/2}e^{Υ(t-t_o)}\partial_{xx} +\lfrac{1}{2}ω^2e^{-Υ(t-t_o)}x^2. This corresponds to a time-dependent mass (TM) Schrödinger equation. We give the specific transformations to i) the more general quadratic (TQ) Schrödinger equation and to ii) a different time-dependent oscillator (TO) equation. For each Schrödinger system, we give the Lie algebra of space-time symmetries, the number states, the coherent states, the squeezed-states and the time-dependent , , (Δx)^2, (Δp)^2, and uncertainty product.
Isolated electrons resting above a helium surface are predicted to have a bound spectrum corresponding to a one-dimensional hydrogen atom. But in fact, the observed spectrum is closer to that of a quantum-defect atom. Such a model is discussed and solved in analytic closed form.
In this paper, we focus on a general class of Schrödinger equations that are time-dependent and quadratic in X and P. We transform Schrödinger equations in this class, via a class of time-dependent mass equations, to a class of solvable time-dependent oscillator equations. This transformation consists of a unitary transformation and a change in the ``time'' variable. We derive mathematical constraints forthe transformation and introduce two examples.
Using the transformations from paper I, we show that the Schrödinger equations for: (1)systems described by quadratic Hamiltonians, (2) systems with time-varying mass, and (3) time-dependent oscillators, all have isomorphic Lie space-time symmetry algebras. The generators of the symmetry algebras are obtained explicitly for each case and sets of number-operator states are constructed. The algebras and the states are used to compute displacement-operator coherent and squeezed states. Some properties of the coherent and squeezed states are calculated. The classical motion of these states is deomonstrated.
A closed form expression for the higher-power coherent states (eigenstates of $a^{j}$) is given. The cases j=3,4 are discussed in detail, including the time-evolution of the probability densities. These are compared to the case j=2, the even- and odd-coherent states. We give the extensions to the "effective" displacement-operator, higher-power squeezed states and to the ladder-operator/minimum-uncertainty, higher-power squeezed states. The properties of all these states are discussed.
We conclude that Murphy's proposal (radiation of the power of the main-bus electrical systems from the rear of the craft) can not explain the anomalous Pioneer acceleration.
We conclude that Katz's proposal (anisotropic heat reflection off of the back of the spacecraft high-gain antennae, the heat coming from the RTGs) does not provide enough power and so can not explain the Pioneer anomaly.
Recently we reported that radio Doppler data generated by NASA's Deep Space Network (DSN) from the Pioneer 10 and 11 spacecraft indicate an apparent anomalous, constant, spacecraft acceleration with a magnitude $\sim 8.5\times 10^{-8}$ cm s$^{-2}$, directed towards the Sun (gr-qc/9808081). Analysis of similar Doppler and ranging data from the Galileo and Ulysses spacecraft yielded ambiguous results for the anomalous acceleration, but it was useful in that it ruled out the possibility of a systematic error in the DSN Doppler system that could easily have been mistaken as a spacecraft acceleration. Here we present some new results, including a critique suggestions that the anomalous acceleration could be caused by collimated thermal emission. Based partially on a further data for the Pioneer 10 orbit determination, the data now spans January 1987 to July 1998, our best estimate of the average Pioneer 10 acceleration directed towards the Sun is $\sim 7.5 \times 10^{-8}$ cm s$^{-2}$.
Radio metric data from the Pioneer 10/11, Galileo, and Ulysses spacecraft indicate an apparent anomalous, constant, acceleration acting on the spacecraft with a magnitude $\sim 8.5\times 10^{-8}$ cm/s$^2$, directed towards the Sun. Two independent codes and physical strategies have been used to analyze the data. A number of potential causes have been ruled out. We discuss future kinematic tests and possible origins of the signal.
The time evolution of even and odd squeezed states, as well as that of squeezed number states, has been given in simple, analytic form. This follows experimental work on trapped ions which has demonstrated even and odd coherent states, number states, and squeezed (but not displaced) ground states. We review this situation and consider the extension to even and odd squeezed number states. Questions of uncertainty relations are also discussed.
The results in the preceding comment are placed on a more general mathematical foundation.
I first review a) the flowering of coherent states in the 1960's, yet b) the discovery of coherent states in 1926, and c) the flowering of squeezed states in the 1970's and 1980's. Then, with the background of the excitement over the then new quantum mechanics, I describe d) the discovery of squeezed states in 1927.
A completely analytic description is given of the motion of a trapped ion which is in either an even or an odd coherent state. Comparison to recent theoretical and experimental work is made.
After beginning with a short historical review of the concept of displaced (coherent) and squeezed states, we discuss previous (often forgotten) work on displaced and squeezed number states. Next, we obtain the most general displaced and squeezed number states. We do this in both the functional and operator (Fock) formalisms, thereby demonstrating the necessary equivalence. We then obtain the time-dependent expectation values, uncertainties, wave-functions, and probability densities. In conclusion, there is a discussion on the possibility of experimentally observing these states.
In this paper we use the Lie algebra of space-time symmetries to construct states which are solutions to the time-dependent Schrödinger equation for systems with potentials $V(x,τ)=g^{(2)}(τ)x^2+g^{(1)}(τ)x +g^{(0)}(τ)$. We describe a set of number-operator eigenstates states, $\{Ψ_n(x,τ)\}$, that form a complete set of states but which, however, are usually not energy eigenstates. From the extremal state, $Ψ_0$, and a displacement squeeze operator derived using the Lie symmetries, we construct squeezed states and compute expectation values for position and momentum as a function of time, $τ$. We prove a general expression for the uncertainty relation for position and momentum in terms of the squeezing parameters. Specific examples, all corresponding to choices of $V(x,τ)$ and having isomorphic Lie algebras, will be dealt with in the following paper (II).
In this article, results from the previous paper (I) are applied to calculations of squeezed states for such well-known systems as the harmonic oscillator, free particle, linear potential, oscillator with a uniform driving force, and repulsive oscillator. For each example, expressions for the expectation values of position and momentum are derived in terms of the initial position and momentum, as well as in the $(α,z)$- and in the $(z,α)$-representations described in I. The dependence of the squeezed-state uncertainty products on the time and on the squeezing parameters are determined for each system.