Uncertainty Relations for a Deformed Oscillator

Abstract

We construct a deformed oscillator whose energy spectra is similar to that of a Morse potential. We obtain a convenient algebraic representation of the displacement and the momentum of a Morse oscillator by expanding them in terms of deformed creation and annihilation operators and we compute their average values between approximate coherent states of the deformed oscillator, and we compare them to the results obtained using the exact Morse coordinate and momenta. Finally we evaluate the temporal evolution of the dispersion (Δx)(Δp) and show that these states are not minimum uncertainty states.

Nonlinear Coherent States and Some of Their Properties

“…We also evaluated the dispersions in the deformed coordinate and momentum between nonlinear coherent states obtained with the displacement operator D D (α). In [22] we evaluated them between Morse-like approximate coherent states obtained as eigenstates of the deformed annihilation operator and found that these approximate coherent states were not minimum uncertainty states though they showed a periodic behaviour with an amplitude near that of a minimum uncertainty state. Figure 4 shows the temporal evolution of the average value of the deformed coordinate D α|x D (t)|α D (upper panel) and in the lower panel we show the corresponding normalized dispersion 2 x D p D /| [x D , p D ] | and the dispersion in the momentum p D for fixed α = 0.5 and a system supporting N = 10 bound states.…”

“…appears in the numerator in one case and in the denominator in the other. It was shown in [22] that the average value of the position and momentum operators pertinent to a Morse-like potential taken between the coherent states given in (25) present a conduct that resembles very closely the classical one. In this section we evaluate the averages of position and momentum for a Morse-like potential between the coherent states obtained via the displacement operator and present a comparison between them.…”

“…where N f is a normalization constant and differ from those obtained by application of the displacement operator (22) mainly in the fact that the factorial f (n)! appears in the numerator in one case and in the denominator in the other.…”

“…This imposes an upper bound on the value of α. Since the average energy of the oscillator is a function of the size α of the coherent state, for a Morselike coherent state there must exist a maximum value of |α| in order to remain in the bound part of the spectrum [22].…”

“…where the expansion coefficients f ij and g ij are known functions of the number operator [22]. Their temporal evolution is calculated taking the averages between coherent states |α(t) D = U(t, t 0 )|α(t 0 ) D with U(t, t 0 ) the time evolution operator, t 0 the initial time.…”