Time dependent classical and quantum oscillators and its applications in mathematical physics

In this study, research was conducted on classical and quantum oscillators, which are an important model in physics and applied sciences to study vibrational motion in nature. In order to consider classical oscillators in more detail, damped harmonic oscillator, forced harmonic oscillator, forced damped harmonic oscillator, random frequency oscillator and random damped harmonic oscillator are examined one by one. Then, Brownian motions, which has gained importance again in recent years, are discussed. First, studies on Brownian motions in the last 20 years were bibliographically reviewed using the Web of Science database. The distribution of studies on Brownian Motions by years, the ranking of the countries in which the studies were conducted, the fields in which more studies were conducted, the journals in which the studies were published, the indexes of the publications published on the subject in the Web of Science index and the number of citations were examined in detail in tables. As a result of these scans, it was realized that the subject became more important as the year passed. Then, the Langevin equation in Brownian motion and the deviation of the Brownian particles from the mean were calculated statistically due to the large number of particles in the motion. In the next step, the deviation amount was calculated with the harmonic oscillator model and compared with the previous result. The result was also analyzed graphically. As a result of the examination, it was seen that the amount of deviation from the mean approached a fixed value. The subject was handled with the Surge Damping equation. The relationship between the force causing the fluctuation and the factor causing the damping was examined. The balance relationship between ripple and damping was emphasized. Finally, the time-varying oscillator model was applied to Brownian motions. It was examined with the help of a mass oscillator model, which increases exponentially over time and is limited to twice the maximum mass value, and how the mean square rate changes over time, and the results are expressed graphically. It was observed that the average frame rate increased over time and approached a constant value. The results obtained were compared with the results in the literature.