Derivation
In the free electron model, the electronic wave function is given by . For a wave packet the group velocity is given by:
Now we can say:
where p is the electron's momentum. Substitute the expression for the group velocity into this last equation and we get:
From this follows the definition of effective mass:
An alternative derivation can be given by considering the Hamiltonian of a free particle and using the de Broglie relation:
The same result is obtained, identifying the Hamiltonian with the total energy:
Read more about this topic: Effective Mass (solid-state Physics)