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).

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

where

For the case of the particle in a one-dimensional box of length L, the potential is ${\displaystyle V_{0}}$ outside the box, and zero for x between ${\displaystyle -L/2}$ and ${\displaystyle L/2}$. The wavefunction is composed of different wavefunctions; depending on whether x is inside or outside of the box, such that: $${\displaystyle \psi ={\begin{cases}\psi _{1},&{\text{if }}x<-L/2{\text{ (the region outside the well)}}\\\psi _{2},&{\text{if }}-L/2<x<L/2{\text{ (the region inside the well)}}\\\psi _{3},&{\text{if }}x>L/2{\text{ (the region outside the well)}}\end{cases}}}$$

For the region inside the box, V(x) = 0 and Equation 1 reduces to $${\displaystyle -{\frac {\hbar ^{2}}{2m}}{\frac {d^{2}\psi _{2}}{dx^{2}}}=E\psi _{2},}$$ resembling the time-independent free schrödinger equation, hence $${\displaystyle E={\frac {k^{2}\hbar ^{2}}{2m}}.}$$ Letting $${\displaystyle k={\frac {\sqrt {2mE}}{\hbar }},}$$ the equation becomes $${\displaystyle {\frac {d^{2}\psi _{2}}{dx^{2}}}=-k^{2}\psi _{2}.}$$ with a general solution of $${\displaystyle \psi _{2}=A\sin(kx)+B\cos(kx)\,.}$$ where A and B can be any complex numbers, and k can be any real number.

For the region outside of the box, since the potential is constant, ${\displaystyle V(x)=V_{0}}$ and equation 1 becomes: $${\displaystyle -{\frac {\hbar ^{2}}{2m}}{\frac {d^{2}\psi _{1}}{dx^{2}}}=(E-V_{0})\psi _{1}}$$

There are two possible families of solutions, depending on whether E is less than ${\displaystyle V_{0}}$ (the particle is in a bound state) or E is greater than ${\displaystyle V_{0}}$ (the particle is in an unbounded state).

If we solve the time-independent Schrödinger equation for an energy ${\displaystyle E>V_{0}}$, letting $${\displaystyle k'={\frac {\sqrt {2m(E-V_{0})}}{\hbar }}}$$ such that $${\displaystyle {\frac {d^{2}\psi _{1}}{dx^{2}}}=-k'^{2}\psi _{1}}$$ then the solution has the same form as the inside-well case: $${\displaystyle \psi _{1}=C\sin(k'x)+D\cos(k'x)}$$ and, hence, will be oscillatory both inside and outside the well. Thus, the solution is never square integrable; that is, it is always a non-normalizable state. This does not mean, however, that it is impossible for a quantum particle to have energy greater than ${\displaystyle V_{0}}$, it merely means that the system has continuous spectrum above ${\displaystyle V_{0}}$, i.e., the non-normalizable states still contribute to the continuous part of the spectrum as generalized eigenfunctions of an unbounded operator.