Editing Quantum well (section)

==== Solutions and energy levels ====
[[File:MCM QW FINWELL.jpg|thumb|A sketch of the bound (<math>E < V_0</math>), quantized energy states for carriers in a finite well model. The wave functions are sinusoidal like but decay exponentially in the barrier region. The energies of these states are less than those in the infinite well model.]]
Using the relevant boundary conditions and the condition that the wave function must be continuous at the edge of the well, we get solutions for the wave vector <math>k</math> that satisfy the following [[transcendental equation]]s:
: <math>\tan\left(\frac{k_nd}{2}\right)=\frac{m_\text{w}^*\kappa}{m_\text{b}^*k_n}\quad\text{(even)}</math>
and
: <math>\tan\left(\frac{k_nd}{2}\right)=-\frac{m_\text{b}^*k_n}{m_\text{w}^*\kappa}\quad\text{(odd)},</math>
where <math>\kappa</math> is the exponential decay constant in the barrier region, which is a measure of how fast the wave function decays to zero in the barrier region. The quantized energy eigenstates inside the well, which depend on the wave vector and the quantum number (<math>n</math>) are given by:
: <math>E_n=\frac{\hbar^2 k_n^2}{2m_\text{w}^*}.</math>

The exponential decay constant <math>\kappa</math> is given by:
: <math>\kappa=\frac{\sqrt{2 m_\text{b}^* (V_0 - E_n)}}{\hbar}</math>

It depends on the eigenstate of a bound carrier <math>E_n</math>, the depth of the well <math>V_0</math>, and the effective mass of the carrier within the barrier region, <math>m^*_\text{b}</math>.

The solutions to the transcendental equations above can easily be found using [[Numerical analysis|numerical]] or graphical methods. There are generally only a few solutions. However, there will always be at least one solution, so one [[bound state]] in the well, regardless of how small the potential is. Similar to the infinite well, the wave functions in the well are sinusoidal-like but exponentially decay in the barrier of the well. This has the effect of reducing the bound energy states of the quantum well compared to the infinite well.<ref name=":0" />