Theoretic impetuses and lengths of Feinberg–Horodecki equation

The exact solution of the Feinberg–Horodecki equation for time-dependent harmonic vector potential has been investigated under a one-dimensional system. The quantized momentum and its corresponding un-normalized wave functions were explicitly obtained. The Fisher information (for time and momentum) and variance (for time and momentum) were calculated using expectation values of time and momentum via Hellman–Feynman theory (HFT). The time and momentum Shannon entropy were obtained using an existing formula. Numerical results were computed for time and momentum Fisher information to confirm the Cramer–Rao inequality. Another numerical results were obtained for time and momentum Shannon entropy to verify Bialynick-Birula, Mycielski (BBM) inequality. The effects of the potential parameters such as mass of the spring and the frequency on the theoretic quantities were fully examined. The new variance inequality was established using the inequalities of Fisher information. The established inequalities were confirmed by numerical results which also satisfied the popular Cramer–Rao inequality. The theoretic impetuses for Fisher information, variance, and Shannon entropy, respectively, were calculated and their variations with some potential parameters were studied.