**4. One-dimensional case of the infinite potential well problem**

The simplest form of the particle in a box model considers a one-dimensional system [1, 2]. Here, the particle may only move backward and forwards along a straight line with impenetrable barriers at either end. The walls of a one-dimensional box may be seen as regions of space with an infinitely large potential energy. Conversely, the interior of the box has a constant zero potential energy. This means that

#### **Figure 3.**

*The barriers outside a one-dimensional box have infinitely large potential, while the interior of the box has a constant zero potential.*

no forces act upon the particle inside the box and it can move freely in that region. However, infinitely large forces repel the particle if it touches the walls of the box, preventing it from escaping. The potential energy in this model is given as:

$$V(\mathfrak{x}) = \begin{cases} 0 & \mathfrak{x}\_{\mathfrak{c}} - \frac{L}{2} < \mathfrak{x} < \mathfrak{x}\_{\mathfrak{c}} + \frac{L}{2} \\ \infty & \text{otherwise} \end{cases}$$

where *L* is the length of the box, *xc* is the location of the center of the box and *x* is the position of the particle within the box. Simple cases include the centered box (*xc* <sup>¼</sup> 0) and the shifted box (*xc* <sup>¼</sup> *<sup>L</sup>* <sup>2</sup>) (**Figure 3**).
