Finite Potential Well - Particle in A 1-dimensional Box

Particle in A 1-dimensional Box

For the 1-dimensional case on the x-axis, the time-independent Schrödinger equation can be written as:

where

,

is Planck's constant,

is the mass of the particle,

is the (complex valued) wavefunction that we want to find,

is a function describing the potential energy at each point x, and

is the energy, a real number, sometimes called eigenenergy.

For the case of the particle in a 1-dimensional box of length L, the potential is zero inside the box, but rises abruptly to a value at x = -L/2 and x = L/2. The wavefunction is considered to be made up of different wavefuctions at different ranges of x, depending on whether x is inside or outside of the box. Therefore the wavefunction is defined such that: