Free-Particle Model:

The free particle model for electron behavior is the simplest which can be considered since it places no energy restrictions on the particle. One begins with the time-independent Schrödinger Equation. Time-dependent wave functions are acceptable to use in this case since they are eigenfunctions of the time-independent hamiltonian:

eq.1

Since no electrostatic potentials exist to influence the electron's behavior, V(r) = 0. Rearranging and using a substitution, k:

eq.2

Solving this homogeneous differential equation produces two linearly independent solutions to Y :

eq.3

This two-parameter solution of can be reduced to one-parameter for simplicity if we assume the electron's momentum vector is directed in only the positive r direction at any values of r and t.

eq.4

Using trigonometric identities to expand this solution:

eq.5

It's obvious that the wave function is a complex function. Complex quantities can be separated into their real and imaginary parts. The real component of Y can be separated:

eq.6

Rewriting using trigonometric identities:

eq.7

The imaginary component of Y can be extracted as well:

eq.8

Rewriting using trigonometric identities:

eq.9

The Free-Particle model is not an entirely realistic representation of electron behavior in the absence of electromagnetic fields. The reason being that the solutions for Y listed above are not square-integrable, one of the three conditions placed on wave functions. Such solutions imply the electron probability density is uniform throughout space, which isnt a realistic description of a particle. The use of wavepackets to realistically describe the behavior of a free electron will be discussed at a later point.