Final answer:
The boundary value problem refers to a mathematical problem in which the solution is sought within a specified domain and subject to certain conditions at the boundaries of the domain. In the case of the particle in a box, the boundary conditions are y(0) = 0 and y(L) = 0. The solution involves finding the eigenvalues and eigenfunctions of the Schrödinger equation.
Step-by-step explanation:
The boundary value problem refers to a mathematical problem in which the solution is sought within a specified domain and subject to certain conditions at the boundaries of the domain. In this case, the boundary conditions are given by y(0) = 0 and y(L) = 0, which means that the solution must vanish at both ends of the domain.
This is a standard problem in quantum mechanics known as the particle in a box. The solution to this problem involves finding the eigenvalues and eigenfunctions of the Schrödinger equation, which describe the energy levels and wave functions of a particle confined to a one-dimensional box.
For a particle in a box, the allowed energy levels are given by E_n = (n^2 * h^2) / (8 * m * L^2), where n is an integer representing the quantum number, h is the Planck's constant, m is the mass of the particle, and L is the length of the box. The corresponding eigenfunction for each energy level is y_n(x) = sqrt(2/L) * sin((n * pi * x) / L), where x is the position within the box.