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

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

van der Spek, Rutger (TU Delft Applied Sciences; TU Delft Electrical Engineering, Mathematics and Computer Science)

Redig, Frank (mentor)
Thijssen, Jos (mentor)

Delft University of Technology

Applied Mathematics | Applied Physics

2019-02-28

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.

Quantum harmonic oscillator
Orsntein-Uhlenbeck process
Coherent States

http://resolver.tudelft.nl/uuid:08ad4a94-483d-46db-8840-6f73c3e48a70

Student theses

bachelor thesis

© 2019 Rutger van der Spek