We introduce the student to the Quantum Mechanic linear potential (constant force) using new variation wave functions for the ground state and the first excited one. This approach use simple wave functions to describe the exact Airy solution.
If the potential is not closely related to a potential that we already know how to solve exactly, the energies and states can still be approximated via the Variational Method. This method is simpler to use than to understand. A simple five-step procedure is given and then illustrated via an example. The principle, in a nutshell, is that if you guess at the ground state and are wrong, the energy you calculate will be higher than the ground state energy, so if you keep guessing and keep getting smaller energies, then you must be getting closer to the ground state wave function. This minimization procedure is called the Variational Method, and it was known prior to quantum mechanics.
Variational methods in quantum mechanics are customarily presented as invaluable techniques to find approximate estimates of ground state energies. In the present paper a short catalogue of different celebrated potential distributions (both 1D and 3D), for which an exact and complete (energy and wavefunction) ground state determination can be achieved in an elementary way, is illustrated. No previous knowledge of calculus of variations is required. Rather, in all presented cases the exact energy functional minimization is achieved by using only a couple of simple mathematical tricks: 'completion of square' and integration by parts. This makes our approach particularly suitable for undergraduates. Moreover, the key role played by particle localization is emphasized through the entire analysis. This gentle introduction to the variational method could also be potentially attractive for more expert students as a possible elementary route toward a rather advanced topic on quantum mechanics: the factorization method. Such an unexpected connection is outlined in the final part of the paper.
In this paper I have drawn out the steps to be followed in order to derive the exact Schr\"odinger wave function for a particle in a general one-dimensional time-dependent linear potential. To this end I have used the so-called Lewis and Riesenfeld invariant method, which is based on finding an exact quantum-mechanical invariant in whose eigenstates the exact quantum states are found. In particular, I have obtained the wave functions of a particle in the linear potential well, driven by a monochromatic electric field.