The document summarizes the Kronig-Penny model, which models an electron in a one-dimensional periodic potential. It describes how the potential is a periodic square wave, allowing the Schrodinger equation to be solved analytically. It then shows the solution of the Schrodinger equation, expressing the eigenfunctions as a linear combination of periodic functions with a periodicity of the potential width. By applying boundary conditions and the translation operator over multiple periods, it derives an expression for the allowed wavevectors and thus the dispersion relation of the model.

1. Chameli Devi Group of Institutions
Title - Kronig’s Penny Model
Submitted by : Piyush Gupta Submitted to : Abhay Tambe

2. INTRODUCTION
• The Kronig – penney model is a simplified model for an electron in a one- dimensional
periodic potential. The possible states that the electron can occupy are determined by the
Schrodinger equation,
•
• In the case of Kronig-Penney model, the potential V(x) is a periodic square wave.
The virtue of this model is that it is possible to analytically determine the energy eigen values
and eigen functions . It is also possible to fine analytic expressions for the dispersion relation
(E vs. k) and the electron density of states.

3. SOLUTION OF THE SCHRODINGER EQUATION FOR THE
KRONIG-PENNY POTENTIAL
• Since the Kronig-Penney potential exhibits translational symmetry, the energy eigenfunctions
of the Schrödinger equation will simultaneously be eigenfunctions of the translation
operator. As we often do in solid state physics, we proceed by seeking the eigenfunctions of
the translation operator. The translation operator T shifts the solutions by one
period, Tψ(x) = ψ(x + a). Notice that any function of the form,
• is an eigenfunction of the translation operator with eigenvalue eika.
• The eigenfunctions of the translation operator can be readily constructed from any two
independent solutions of the one-dimensional Schrödinger equation. A convenient choice is,

4. The solutions in region 1 (0 < x < b) are,
while the solutions in region 2 (b < x < a) are,
Here,
For energies where k1 or k2 are imaginary, the solutions are still real since cos(iθ) = cosh(θ) and
sin(iθ) = isinh(θ).

5. • Any other solution can be written as a linear combination of ψ1(x) and ψ2(x). In particular, ψ1(x + a) and
ψ2(x + a) can be written in terms of ψ1(x) and ψ2(x). These solutions are related to each other by the
matrix representation of the translation operator.
The elements of the translation matrix can be determined
by evaluating the equation above and its derivative
at x = 0.
The eigenfunctions and eigenvalues λ of this 2 × 2 matrix
are easily determined to be,
where
and

6. • If periodic boundary conditions are used for a potential with N unit cells, then applying the translation
operator N times brings the function back to its original position,
•
The eigenvalues of the translation operator are therefore
the solutions to the equation λN = 1. These solutions are,
where j is an integer between -N/2 and N/2, L = Na is the length of the crystal, and kj = 2πj/L are the
allowed k values in the first Brillouin zone. The dispersion relation can be determined by first calculating α for a
specific energy, solving for the eigenvalues λ and then solving the equation above for the wavenumber k,
Whether the eigenvalues are real or imaginary depends on the magnitude of α. If α² > 4, the eigenvalues will be real
and the solutions fall in a forbidden energy gap. If α² < 4, the eigenvalues will be a complex conjugate pair
λ+ = eika and λ- = e-ika.