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 $e^{i p_0 x/\hbar} e^{- (1 \mp i \delta) (x - x_0)^2 / 4 (\Delta x_0)^2}$, where $\delta = \sqrt{(2\Delta x_0 \Delta p_0/\hbar)^2 -1}$ denotes a measure of deviation from the minimum uncertainty or the initial position-momentum correlation $\delta = 2\Delta (xp)_0 / \hbar$. The travelling Gaussian wave packet takes, up to a time-dependent phase and normalization factor, the form $e^{i p_c x/\hbar} e^{- (1 - 2 i \Delta (xp)_t/\hbar) (x - x_c)^2 / 4 (\Delta x_t)^2}$ 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.