  One-dimensional particle states are constructed according to orthogonality
conditions, without requiring boundary conditions. Free particle states are
constructed using Dirac's delta function orthogonality conditions. The states
(doublets) depend on two quantum numbers: energy and parity. With the aid of
projection operators the particles are confined to a constrained region, in a
way similar to the action of an infinite well potential. From the resulting
overcomplete basis only the mutually orthogonal states are selected. Four
solutions are found, corresponding to different non-commuting Hamiltonians.
Their energy eigenstates are labeled with the main quantum number n and parity
"+" or "-". The energy eigenvalues are functions of n only. The four cases
correspond to different boundary conditions: (I) the wave function vanishes on
the boundary, (II) the derivative of the wavefunction vanishes on the
boundary,(III) periodic (symmetric) boundary conditions, (IV) periodic
(antisymmetric)boundary conditions . Among the four cases, only solution (III)
forms a complete basis in the sense that any function in the constrained
region, can be expanded with it. By extending the boundaries of the constrained
region to infinity, only solution (III) converges uniformly to the free
particle states. Orthogonality seems to be a more basic requirement than
boundary conditions. By using projection operators, confinement of the particle
to a definite region can be achieved in a conceptually simple and unambiguous
way, and physical operators can be written so that they act only in the
confined region.
