The finite square well and the tunneling barrier are two closely related quantum-mechanics models. Both arise from the
time-independent Schrödinger equation, but unlike the infinite well, the potential is not infinitely large outside the
allowed region. Because of that, the wave function does not stop abruptly at the edge of the well or barrier. Instead,
it penetrates into classically forbidden regions and decays exponentially there. This is one of the clearest signatures
of quantum behavior.
In a finite potential well, a particle can have bound states with discrete energies, but only if those energies lie
below the well depth. The allowed energies are found by matching the wave function and its derivative at the edges of
the well. For a symmetric well, this produces transcendental equations rather than a simple closed formula. The exact
bound-state energies therefore usually require numerical solution.
Characteristic wave numbers in a finite well.
\[
k=\frac{\sqrt{2mE}}{\hbar},
\qquad
\alpha=\frac{\sqrt{2m(V_0-E)}}{\hbar}
\]
Here, \(m\) is the particle mass, \(E\) is the bound-state energy measured relative to the bottom of the well, \(V_0\)
is the well depth, and \(\hbar\) is the reduced Planck constant. Inside the well, the solution oscillates with wave
number \(k\). Outside the well, where the energy is below the potential, the wave function decays with constant
\(\alpha\).
For a symmetric finite well of half-width \(a\), the exact conditions for bound states are commonly written as
Even and odd bound-state conditions.
\[
k\tan(ka)=\alpha
\qquad \text{(even states)}
\]
\[
-k\cot(ka)=\alpha
\qquad \text{(odd states)}
\]
These equations do not usually simplify into a direct algebraic expression for \(E\), which is why the finite-well
part of this topic is usually solved numerically or approximately. Still, the physical interpretation is clear:
stronger and wider wells can support more bound states, while shallow or narrow wells may support only one or even no
bound state depending on the parameters.
Tunneling through a rectangular barrier
The second part of this topic concerns a particle approaching a barrier of height \(V_0\) and thickness \(L\). In
classical mechanics, if the particle energy \(E\) is smaller than \(V_0\), the particle cannot cross the barrier.
Quantum mechanics predicts something different: the wave function penetrates the barrier and there is a nonzero
probability that the particle appears on the other side. This is quantum tunneling.
For a sufficiently thick barrier and for \(E < V_0\), a commonly used approximation for the transmission probability is
Approximate tunneling transmission.
\[
T \approx \frac{16E(V_0-E)}{V_0^2}e^{-2\kappa L}
\]
The exponential-decay constant inside the barrier is
Barrier decay constant.
\[
\kappa=\frac{\sqrt{2m(V_0-E)}}{\hbar}
\]
The crucial factor is the exponential term \(e^{-2\kappa L}\). Even modest increases in barrier width \(L\), barrier
height \(V_0\), or particle mass \(m\) can reduce the transmission probability dramatically. This is why tunneling is
especially significant for very light particles such as electrons and for very thin barriers.
Sample tunneling estimate
Consider an electron incident on a barrier with \(V_0=10\ \mathrm{eV}\), \(E=5\ \mathrm{eV}\), and thickness
\(L=0.2\ \mathrm{nm}\). Since \(E < V_0\), the barrier region is classically forbidden. First compute
\[
V_0-E = 5\ \mathrm{eV}.
\]
Then
\[
\kappa=\frac{\sqrt{2m(V_0-E)}}{\hbar}.
\]
Because the electron mass is small but the barrier is still several tenths of a nanometer wide, the factor
\(e^{-2\kappa L}\) becomes strongly suppressive. As a result, the transmission probability is very small. This is why
the output is often described qualitatively as “exponentially small” unless the barrier is extremely thin or the
particle energy is very close to the barrier height.
Physical meaning
Finite wells explain why bound states in real materials are not perfectly confined: the wave function leaks slightly
outside the classically allowed region. Tunneling explains phenomena such as alpha decay, scanning tunneling microscopy,
Josephson junction behavior, field emission, and electron transport in nanoscale devices. In all of these cases, the
wave function extends into forbidden regions and produces observable effects that classical mechanics cannot reproduce.
At a more advanced university level, one usually solves the finite-well bound-state equations numerically and uses the
exact transfer-matrix or matching-condition formulas for barrier transmission rather than only the thick-barrier
approximation. Even so, the approximate formula given here is extremely useful for developing intuition: tunneling is
controlled primarily by exponential decay inside the barrier.
| Concept |
Main relation |
Meaning |
| Oscillatory wave number |
\(k=\sqrt{2mE}/\hbar\) |
Behavior inside the classically allowed region |
| Decay constant |
\(\alpha=\sqrt{2m(V_0-E)}/\hbar\) |
Exponential decay outside a finite well |
| Even bound states |
\(k\tan(ka)=\alpha\) |
Finite-well energy condition for symmetric states |
| Odd bound states |
\(-k\cot(ka)=\alpha\) |
Finite-well energy condition for antisymmetric states |
| Tunneling decay constant |
\(\kappa=\sqrt{2m(V_0-E)}/\hbar\) |
Controls penetration into a barrier |
| Approximate transmission |
\(T \approx \dfrac{16E(V_0-E)}{V_0^2}e^{-2\kappa L}\) |
Thick-barrier tunneling probability for \(E
|