Editing Wave function (section)

===Finite potential barrier===
[[File:Finitepot.png|thumb|Scattering at a finite potential barrier of height {{math|''V''<sub>0</sub>}}. The amplitudes and direction of left and right moving waves are indicated. In red, those waves used for the derivation of the reflection and transmission amplitude. {{math|''E'' > ''V''<sub>0</sub>}} for this illustration.]]

One of the most prominent features of wave mechanics is the possibility for a particle to reach a location with a prohibitive (in classical mechanics) [[potential energy|force potential]]. A common model is the "[[potential barrier]]", the one-dimensional case has the potential
<math display="block">V(x)=\begin{cases}V_0 & |x|<a \\ 0 & | x | \geq a\end{cases}</math>
and the steady-state solutions to the wave equation have the form (for some constants {{math|''k'', ''κ''}})
<math display="block">\Psi (x) = \begin{cases}
A_{\mathrm{r}}e^{ikx}+A_{\mathrm{l}}e^{-ikx} & x<-a, \\
B_{\mathrm{r}}e^{\kappa x}+B_{\mathrm{l}}e^{-\kappa x} & |x|\le a, \\
C_{\mathrm{r}}e^{ikx}+C_{\mathrm{l}}e^{-ikx} & x>a.
\end{cases}</math>

Note that these wave functions are not normalized; see [[scattering theory]] for discussion.

The standard interpretation of this is as a stream of particles being fired at the step from the left (the direction of negative {{mvar|x}}): setting {{math|1=''A''<sub>r</sub> = 1}} corresponds to firing particles singly; the terms containing {{math|''A''<sub>r</sub>}} and {{math|''C''<sub>r</sub>}} signify motion to the right, while {{math|''A''<sub>l</sub>}} and {{math|''C''<sub>l</sub>}} – to the left. Under this beam interpretation, put {{math|1=''C''<sub>l</sub> = 0}} since no particles are coming from the right. By applying the continuity of wave functions and their derivatives at the boundaries, it is hence possible to determine the constants above.

[[File:Quantum dot.png|thumb|upright=1.3|3D confined electron wave functions in a quantum dot. Here, rectangular and triangular-shaped quantum dots are shown. Energy states in rectangular dots are more ''s-type'' and ''p-type''. However, in a triangular dot the wave functions are mixed due to confinement symmetry. (Click for animation)|link=File:QuantumDot_wf.gif]]

In a semiconductor [[crystallite]] whose radius is smaller than the size of its [[exciton]] [[Bohr radius]], the excitons are squeezed, leading to [[Potential well#Quantum confinement|quantum confinement]]. The energy levels can then be modeled using the [[particle in a box]] model in which the energy of different states is dependent on the length of the box.