Graphical Representation of the Quantum Mechanical Two-Dimensional Harmonic Oscillator Wavefunction

Fan Hong-Yi

Charles W. Struck, William H. Fonger

W. Blake Laing

We generalize the harmonic-order N-body DPT method for an isotropically confined quantum system of identical interacting bosons to first-anharmonic order and, in principle, higher orders. We introduce a graphical decomposition of the perturbative expansion of the N-body Hamiltonian. We calculate the Clebsch-Gordon coefficient tensors that couple together 3 irreducible representations of SN analytically, and analytically transformed the graphical basis to collective coordinates. We calculate the N-body wavefunction and density profile in general and have demonstrated agreement with an analytic model. We apply this formalism to the exactly-solvable example of a trapped gas of atoms interacting with a (fully-interacting) harmonic-oscillator "Hooke's law" interaction and compare with the dimensional expansion of the exact ground-state wavefunction and density profile. We report progress on the example of a cold gas BEC (with zero angular momentum).

Joseph H. Simmons, Kelly S. Potter

Joseph R. Noonan, Maaz ur Rehman Shah, Luogen Xu +1 more

The factorization method of Schrödinger shows us how to determine the energy eigenstates without needing to determine the wavefunctions in position or momentum space. A strategy to convert the energy eigenstates to wavefunctions is well known for the one-dimensional simple harmonic oscillator by employing the Rodrigues formula for the Hermite polynomials in position or momentum space. In this work, we illustrate how to generalize this approach in a representation-independent fashion to find the wavefunctions of other problems in quantum mechanics that can be solved by the factorization method. We examine three problems in detail: (i) the one-dimensional simple harmonic oscillator; (ii) the three-dimensional isotropic harmonic oscillator; and (iii) the three-dimensional Coulomb problem. This approach can be used in either undergraduate or graduate classes in quantum mechanics.

Mai-Lin Liang, Fulin Zhang

After a unitary transformation, exact wavefunctions of the time-dependent two-dimensional harmonic oscillator in a time-dependent magnetic field are written in the form of the product of two one-dimensional time-dependent harmonic oscillators in the ( x , p y ) representation. Applying the Heisenberg correspondence principle (HCP) to the system, the exact classical solution is obtained from the quantum matrix elements. Meanwhile, the coherent state is discussed.