Understanding the relationship between k (i. e. momentum) and e of an electron living in a solid is important towards understanding the behavior and properties of semiconductor materials. appreciating the parabolic relationship between k and e for a truly “free” electron is a fundamental relationship to which electron energy / momentum discussions are referenced in “real” materials.
derive the relationship between e and k for an electron in an infinitely long (l = infinity), 1d box. think of this as a truly “free” electron in 1d space. in this approximation, the electron is not quantized as in a real-life potential well or macroscopic materials, i. e. the spacing between permissible k values is 1/infinity = 0, i. e. k becomes continuous.