The Exact Solutions in Quantum Mechanics

In this study we consider two different and particular quantum systems. The first system is a one-dimensional particle confined within an infinite well with the left wall at rest and the right wall moving with or without acceleration. This is a quantum configuration with time-dependent Hamiltonian where the conservation of energy does not hold anymore. Our task in this problem is to solve the correspondence time dependent Schrödinger equation and find the energy eigenvalues and eigenfunctions. The second system is a particle which undergoes a modified radial attraction force in two dimensions. The modification is such that the corresponding potential depends on not only the radial coordinate but also the polar angle. We find the energy spectrum of the particle and using the concept of coherent states, we show that the classical trajectory of a classical particle directed by a similar radial potential is almost the same as the predicted probability density of the quantum particle by the coherent states. Keywords: One-dimensional infinite well, Time-dependent potential, Exact solution, Coherent state.