Final answer:
Due to the complexity involved in non-periodic systems, band structures are typically obtained through numerical methods rather than analytic diagonalization. Eigenvalues are determined by numerically solving the Schrödinger's equation, with similarities drawn from simpler systems like the 'particle in a box' to understand the quantum mechanics at play.
Step-by-step explanation:
For open boundary conditions, crystal momentum is indeed not a good quantum number, meaning that band structure cannot be defined in the same way as in a system with periodic boundary conditions. To obtain the eigenvalues of the system, we need to solve the Schrödinger's equation for the crystal's potential, typically not analytically solvable. What can be done are numerical diagonalizations of the Hamiltonian matrix in the position basis. Band theory of solids often involves complex quantum mechanical calculations that can provide insights into the energy levels of electrons.
In particular, when there are no periodic boundary conditions (as with large systems or surfaces), the problem often must be handled numerically. Wavefunctions are calculated by making use of the properties of particles confined in a potential well, drawing on analogies from simpler quantum mechanical systems such as the particle in a box model. Perturbation methods or variational principles may also be applied in some cases, but these typically yield approximate solutions.
Analytical solutions for the band structure in the absence of periodic boundary conditions are not generally available due to the complexity of the potential landscape in real materials. As a result, computer simulations and numerical methods are widely used in the field of condensed matter physics to study these systems.