The finite potential well (also known as the finite square well) is a concept from quantum mechanics. It is an extension of the infinite potential well, in which a particle is confined to a "box", but one which has finite potential "walls". Unlike the infinite potential well, there is a probability associated with the particle being found outside the box. The quantum mechanical interpretation is unlike the classical interpretation, where if the total energy of the particle is less than the potential energy barrier of the walls it cannot be found outside the box. In the quantum interpretation, there is a non-zero probability of the particle being outside the box even when the energy of the particle is less than the potential energy barrier of the walls (cf quantum tunnelling). 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 the reduced Planck's constant, 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 outside the box, and zero for x between and . The wavefunction is considered to be made up of different wavefunctions at different ranges of x, depending on whether x is inside or outside of the box. Therefore, the wavefunction is defined such that: Inside the box For the region inside the box, V(x) = 0 and Equation 1 reduces to Letting the equation becomes This is a well-studied differential equation and eigenvalue problem with a general solution of Hence, Here, A and B can be any complex numbers, and k can be any real number. Outside the box For the region outside of the box, since the potential is constant, and equation becomes: There are two possible families of solutions, depending on whether E is less than (the particle is bound in the potential) or E is greater than (the particle is free). For a free particle, , and letting produces with the same solution form as the inside-well case: This analysis will focus on the bound state, where . Letting produces where the general solution is exponential: Similarly, for the other region outside the box: Now