Coherent-Squeezed State Representation of Travelling General Gaussian Wave Packets

Abstract

Using the time-dependent annihilation and creation operators, the invariant operators, for a free mass and an oscillator, we find the coherent-squeezed state representation of a travelling general Gaussian wave packet with initial expectation values, $x_0$ and $p_0$, of the position and momentum and variances, $\Delta x_0$ and $\Delta p_0$. The initial general Gaussian wave packet takes, up to a normalization factor, the form $e^{i{p}_{0}x/\hslash }{e}^{-(1\mp i\delta )(x-x_0{)}^2/4(\Delta x_0{)}^2}$, where $\delta =\sqrt{(2\Delta x_0\Delta p_0/\hslash {)}^2-1}$ denotes a measure of deviation from the minimum uncertainty or the initial position-momentum correlation $\delta =2\Delta (xp{)}_0/\hslash$. The travelling Gaussian wave packet takes, up to a time-dependent phase and normalization factor, the form $e^{i{p}_{c}x/\hslash }{e}^{-(1-2i\Delta (xp{)}_t/\hslash )(x-x_c{)}^2/4(\Delta x_t{)}^2}$ and the centroid follows the the classical trajectory with $x_c(t)$ and $p_c(t)$. The position variance is found to have additionally a linearly time-dependent term proportional to $\delta$ with both positive and negative signs.

RevTex 14 pages, no figure