Continuity conditions

The Schrödinger equation requires calculating the second derivative of the wave function with respect to position. Therefore, at the very least the wave function and its first derivative must be continuous.

1. Wave function continuous at $x = \dfrac{a}{2}$ :

$B \cos\biggl(\dfrac{ka}{2}\biggr) = C \text{exp}\biggl(-\dfrac{ga}{2}\biggr) \quad \text{(1)}$

2. Derivative of the wave function at $x = \dfrac{a}{2}$ :

$- B k \sin\biggl(\dfrac{ka}{2}\biggr) = - C g \text{exp}\biggl(-\dfrac{ga}{2}\biggr) \quad \text{(2)}$

Substituting the two equations into one another gives:

$- B k \sin\biggl(\dfrac{ka}{2}\biggr) = - g B \cos\biggl(\dfrac{ka}{2}\biggr) \quad \text{or}$

$k \cdot \tan\biggl(\dfrac{ka}{2}\biggr) = \gamma = \sqrt{\dfrac{2mV_0}{\hbar^2} - k^2}$

This condition must be satisfied for a solution of the homogeneous system of equations (1), (2) to exist. The last equation is a transcendental equation for determining the possible values of $k$ , and hence the possible values of the energy $E$ , which, as is well known, cannot generally be solved in closed form.

Back