KAHIBARO
Table of Contents
Beyond the Infinite Well
In the idealized particle in a box, the walls are infinitely high, so the particle can never be found outside the box. A finite potential well is more realistic. The walls have a limited height, which means the particle is mostly confined, but not perfectly confined. This changes the allowed energies and leads to one of the most important quantum ideas, the possibility that the particle can exist slightly inside regions that would be forbidden in classical physics.
A finite potential well is a region where the potential energy is lower than in the surrounding space. In one dimension, a common model is
$$
V(x) =
\begin{cases}
0, & |x| > a \\
- V_0, & |x| \le a
\end{cases}
$$
where $V_0 > 0$ is the depth of the well, and the well extends from $x = -a$ to $x = a$.
Classical and Quantum Pictures
Classically, if a particle has total energy less than the height of the outside region, it would remain trapped inside the well and could never enter the outside region.
Quantum mechanically, the wave function does not stop sharply at the boundaries. Instead, it extends a little into the outside region. This means there is a nonzero probability of finding the particle outside the well, even when its energy is less than the outside potential level.
This is different from a free particle and different from the infinite well. In the finite well, confinement is incomplete.
For a finite well, the wave function must remain continuous, and its slope must also be continuous at each boundary, provided the potential does not become infinite there.
Bound States
The most important states in a finite well are bound states. These occur when the total energy is less than the outside potential value. In the potential written above, the outside potential is $0$, so bound states satisfy
$$
- V_0 < E < 0
$$
Inside the well, the particle behaves like a wave. Outside the well, the wave function decreases exponentially rather than oscillating.
This means the particle is localized mainly in the well, but with tails extending beyond the edges.
Shape of the Wave Function
To understand the wave function, divide space into three regions.
Inside the well, where $V(x) = -V_0$, the time independent Schrödinger equation gives oscillatory solutions. Outside the well, where $V(x) = 0$ and $E < 0$, the solutions are exponential.
A typical form is
$$
\psi(x) =
\begin{cases}
A e^{\kappa x}, & x < -a \\
B \cos(kx) + C \sin(kx), & |x| \le a \\
D e^{-\kappa x}, & x > a
\end{cases}
$$
where
$$
k = \frac{\sqrt{2m(E + V_0)}}{\hbar},
\qquad
\kappa = \frac{\sqrt{-2mE}}{\hbar}
$$
The oscillating part inside the well shows wave behavior in the classically allowed region. The decaying exponential outside the well shows penetration into the classically forbidden region.
Inside the finite well, bound state wave functions oscillate.
Outside the well, bound state wave functions decay exponentially.
They do not remain exactly zero outside the well.
Even and Odd States
Because the well is symmetric about $x = 0$, the bound state wave functions can be chosen to have definite symmetry.
An even state satisfies
$$
\psi(-x) = \psi(x)
$$
and an odd state satisfies
$$
\psi(-x) = -\psi(x)
$$
This simplifies the mathematics. For even states, the inside solution can be written with cosine. For odd states, it can be written with sine.
For even states,
$$
\psi(x) =
\begin{cases}
A e^{\kappa x}, & x < -a \\
B \cos(kx), & |x| \le a \\
A e^{-\kappa x}, & x > a
\end{cases}
$$
For odd states,
$$
\psi(x) =
\begin{cases}
A e^{\kappa x}, & x < -a \\
B \sin(kx), & |x| \le a \\
- A e^{-\kappa x}, & x > a
\end{cases}
$$
The boundary conditions lead to special equations that determine which energies are allowed.
Energy Quantization
Just like in the infinite well, not every energy is allowed. The need to match the wave function smoothly at the boundaries selects only certain energies.
For even states, the condition is
$$
k \tan(ka) = \kappa
$$
For odd states, the condition is
$$
- k \cot(ka) = \kappa
$$
These equations are called transcendental equations because the energy appears inside trigonometric functions and square roots. In general, they are solved graphically or numerically.
Allowed bound state energies in a finite well are still quantized, but they are not given by a simple formula like the infinite square well.
They are found from
$$
k \tan(ka) = \kappa
\quad \text{even states}
$$
and
$$
- k \cot(ka) = \kappa
\quad \text{odd states}
$$
Comparison with the Infinite Well
The finite well and infinite well are similar because both produce discrete bound energies. But the finite well has important differences.
| Feature | Infinite well | Finite well |
|---|---|---|
| Wave function outside well | Exactly zero | Exponentially decaying |
| Number of bound states | Infinite | Finite |
| Energy formula | Simple explicit formula | Found from matching conditions |
| Confinement | Perfect | Incomplete |
Because the walls are not infinitely high, only a limited number of bound states can fit into the well. A deeper or wider well can hold more bound states.
Number of Bound States
A finite well does not support infinitely many bound states. If the well is shallow or narrow, there may be only one or a few.
This is physically reasonable. Weak confinement cannot trap arbitrarily high energy states. States with too much energy are no longer bound and become unbound states.
In one dimension, an attractive finite well always supports at least one bound state. This is a special and important result.
Unbound States
If the particle energy is greater than or equal to the outside potential level, then the particle is not bound. In the example above, that means
$$
E \ge 0
$$
In that case, the wave function oscillates outside the well as well as inside it. These are scattering states rather than bound states. The particle is not confined to the well.
The treatment of scattering is broader than this chapter, but it is useful to know that a finite well has both discrete bound states and continuous unbound states.
Penetration Depth
The exponential tail outside the well has a characteristic length scale. Since the wave function behaves like $e^{-\kappa x}$, the decay length is roughly
$$
\frac{1}{\kappa} = \frac{\hbar}{\sqrt{-2mE}}
$$
A smaller value of $\kappa$ means deeper penetration outside the well. If the bound energy is close to the top of the well, the tail extends farther.
This penetration has no classical analog. It is a direct consequence of the wave nature of matter.
A finite potential barrier or wall does not stop the wave function abruptly.
Instead, the wave function penetrates into the classically forbidden region over a distance of order
$$
\frac{1}{\kappa}
$$
Physical Meaning
Finite wells are used to model many real systems. Electrons in atoms, charge carriers in semiconductor structures, and particles in molecular binding can all be approximated using finite confinement rather than perfectly rigid walls.
The finite well teaches an important lesson. Quantum confinement is not absolute unless the potential is infinite. Real physical boundaries usually allow some leakage of the wave function.
Visualizing the Finite Well
The picture below shows the potential and a typical bound state wave function.
Main Ideas to Remember
A finite potential well confines a particle with a potential that is lower inside than outside. Bound states have discrete energies, but unlike the infinite well, the wave function extends outside the well and decays exponentially. The energies are found by applying continuity conditions at the boundaries, leading to separate equations for even and odd states. Only a finite number of bound states exist, and this makes the finite well a more realistic model of physical confinement.
Key facts for finite potential wells:
$$
- V_0 < E < 0 \quad \text{for bound states}
$$
Inside the well, $\psi$ oscillates.
Outside the well, $\psi$ decays exponentially.
Allowed energies are quantized and found from boundary matching.
A finite well supports only a finite number of bound states.