On quantum analogue of dynamical stabilisation of inverted harmonic oscillator by time periodical uniform field
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Quantum analogue of stabilised forced oscillations around an unstable equilibrium position is explored by solving the non-stationary Schrödinger equation (NSE) of the inverted harmonic oscillator (IHO) driven periodically by spatial uniform field of frequency $\Omega$, amplitude $F_{0}$ and phase $\phi$, i.e. the system with the Hamiltonian of $\hat{H} = (\hat{p}^{2}/2m) − (m\omega^{2}x^{2}/2) − F_{0}x sin(\Omega t + \phi)$. The NSE has been solved both analytically and numerically by Maple 15 in dimensionless variables $\xi = x \sqrt{m\omega/hbar}, f_{0} = F_{0}/\omega\sqrt{hbarm\omega}$ and $\tau = \omega t$. The initial condition (IC) has been specified by the wave function (w.f.) of a generalised Gaussian type which suits well the corresponding quantum IC operator.The solution obtained demonstrates the non-monotonous behaviour of the coordinate spreading $\sigma (\tau) = \sqrt{\overline{(\Delta\xi^{2}(\tau))}}$ which decreases first from quite macroscopic values of $\sigma_{0} = 2^{12,...,25}$ to minimal one of $\sim(1/\sqrt{2})$ at times $\tau$ < $\tau_{0} = 0.125 1n(16\sigma^{4}_{0} + 1)$ and then grows back unlimitedly. For certain phases $\phi$ depending on the $\Omega/\omega$ ratio and $n = log_{2}\sigma_{0}$, the mass centre of the packet $\xi_{av}(\tau) = \overline{\hat{x}(\tau)} · \sqrt{m\omega/hbar}$ delays approximately two natural ‘periods’ $\sim(4\pi/\omega)$ in the area of the stationary point and then escapes to ‘+’ or ‘−’ infinity in a bifurcating way. For ‘resonant’ $\Omega = \omega$, the bifurcation phases $\phi$ fit well with the regression formula of Fermi–Dirac type of argument n with their asymptotic $\phi(\Omega, n\rightarrow\infty)$ obeying the classical formula $\phi_{cl}(\Omega) = −arctg \Omega$ for initial energy $E = 0$ in the wide range of $\Omega = 2^{−4}, ..., 2^{7}$.
VIKTOR V CHISTYAKOV^{1} ^{}
Volume 95, 2021
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