Two Particles in the Harmonic Oscillator Potential

S. Brandt, Hans Dieter Dahmen, Tilo Stroh

Steven M. Pascal

Another "simple" potential is one experienced by a spring, the simple harmonic oscillator potential (SHO). To understand how a quantum particle behaves in such a potential, first we study how a classical particle, such as a mass on a spring, behaves. Then we change to using the TISE with the above potential. Solving this TISE is not a simple task. One way of solving a differential equation is to see if someone else has already solved it. Fortunately, a French mathematician named Hermite solved a related equation in the 19 th century. His solutions can be multiplied by a simple exponential function, and these modified Hermite polynomials can be shown to solve the TISE for the SHO. Interestingly, the resulting trapped particles have quantized energies that are evenly spaced. It is for this reason that bond vibrations, which can be modeled as springs on a mass, are found to be quantized in nearly evenly spaced energy levels, at least for relatively small vibrations. The SHO TISE also provides an opportunity to introduce two new operators: the creation and annihilation operators.

Yin Xiang-yu

To study the rotating trap which is related with BEC,two-dimensional harmonic oscillator is ana- lyzed and Hamihonian is provided.Wave function and energy eigenvalue are given.A coincidence is found that the rotating trap of two-dimensional harmonic oscillator has the similar form as the potential of charged particle in mag- netic field.

S. Brandt, Hans Dieter Dahmen

Tian-Hai Zeng, Zhaobin Liu, Kai Li +2 more

Isolated coupled-harmonic-oscillator here is the system of two distinguishable particles coupled with a harmonic oscillator interaction potential. Each particle stays in a mixed state due to entanglement. However, in center-of-mass reference frame, we obtain quasi wavefunction of the first particle expressing quasi pure state by replacing the second coordinate in the total wavefunction. We discuss the similar systems with the first particle and the potential being same and the second mass changing from micro to macro one. Measured by fidelity and coherence, the quasi pure state approaches to the pure state of a usual harmonic oscillator with same mass and similar potential. It conversely shows that the latter purely superposed state in position representation and its coherence originate from those of the first particle, which are related with some neglected macro object and the interaction between them. The current results provide a possible clue to new insights into quantum states.

S. Brandt, Hans Dieter Dahmen