Final answer:
The Kronig-Penny model is used to describe particles in a periodic potential. To model an anti-symmetric solution, a piecewise wave function is constructed with a periodicity and a potential function that is zero except for a specific region. The wave function must be symmetric about x = 0 and normalizable to ensure finite probability density.
Step-by-step explanation:
The Kronig-Penny model is used to describe the behavior of particles in a periodic potential. To model an anti-symmetric solution, we can use a piecewise wave function with periodicity a and a potential function that is zero except for a region of width w. By placing one edge of the barrier at the origin, x = 0, the algebra becomes easier.
Anti-symmetric solutions are odd functions about x = 0. In region I (x < 0), the wave function, denoted as w₁(x), can be described by the stationary Schrödinger equation. In region II (0 ≤ x ≤ L), the wave function y(x) satisfies the same equation, and in region III (x > L), y(x) is the solution. These three regions define the piecewise wave function.
By solving the Schrödinger equation for the periodic potential, we can find the allowed energies and their corresponding wave functions. The wave functions must be symmetric about x = 0 and normalizable. This ensures that the probability density is finite when integrated.