From the Quantum Harmonic Oscillator to the Orstein-Uhlenbeck Process and Back

Abstract

In this thesis, the relation between the generator of the OrnsteinUhlenbeck process and the Hamiltonian of the quantum harmonic oscillator is used to derive a new understanding of the evolution of certain quantum states. More precisely, we transform the Hamiltonian with respect to the ground state and corresponding eigenvalue to find that it is equal to minus the generator of the Ornstein-Uhlenbeck process. Next, we use the knowledge of the evolution of distributions in the OrnsteinUhlenbeck process to obtain the time evolution of corresponding quantum states. Specifically, we derived that the evolution of normal distributions in the Ornstein-Uhlenbeck process remain normally distributed with varying mean and variance. Furthermore, the ground state of the harmonic oscillator is equal to the square root of the reversible distribution of the Ornstein-Uhlenbeck process. Combining these results gives us the evolution of quantum states with an almost Gaussian wave function. If we confine one degree of freedom in the end result, we obtain the coherent states of the quantum harmonic oscillator. These are Gaussian wave packets, which means that the probability density is Gaussian with constant variance and oscillating mean. Coherent states most closely resemble classical particles in the harmonic oscillator and minimise Heisenberg’s uncertainty principle.