Potential Well

Bound states and discrete energies in a confining region. Theory, formulas, worked examples and interactive graphs.

Conceptual context

A potential well is a region where a particle is confined because the potential outside is higher. In classical physics, a trapped particle could have almost any energy compatible with staying inside. In quantum mechanics the particle is described by a wave, and that wave cannot have an arbitrary shape inside the confinement. The $L$ acts as a spatial ruler: only certain standing patterns fit. Therefore the system has discrete values of $E_n$, labeled by the $n$. The potential well turns available space into an energy filter.

🟢 Essential level

The central idea is that a confined particle is not a tiny bead freely bouncing between walls, but a quantum wave forced to fit the available space.

If the $L$ is small, the wave must bend more sharply, and that bending corresponds to a higher $E_n$. If the $L$ grows, longer waves fit and the lowest energy decreases. The $n$ labels the allowed modes: the first mode contains half a wave inside the well, the second contains more oscillation, and so on. The $\lambda$ is not chosen freely; it is imposed by boundary conditions. That is why the well produces separated levels rather than a continuous energy ladder. The quantum box turns space into selection, not mere enclosure. It is like tuning a guitar string by cutting it to different lengths: a shorter string only allows higher-pitched, well-defined notes, and it can never sound at any arbitrary in-between pitch you might wish to request.

🔵 Formal level

Basic formulas

In the one-dimensional infinite well, the walls require the wavefunction to vanish at the edges. That boundary condition selects only standing wavelengths compatible with the $L$ and a positive integer $n$.

Allowed wavelength in the well

The allowed $\lambda$ is therefore tied to the geometry of confinement. Combining that wave with quantum momentum and nonrelativistic kinetic energy gives the discrete stationary-state energy.

Infinite well energy

This relation uses $E_n$, $n$, $L$, $m$, and $\hbar$. It shows that energy increases with the square of $n$ and decreases with the square of $L$. The formal reading also requires checking that n is a positive integer and that the width is expressed in meters before substitution.

Real student questions

Why can the particle not rest with zero energy? Because a completely flat wave cannot satisfy the nodes imposed by the walls.

Is $n$ measured directly? Usually it is inferred from energy, nodes, or transitions between states.

Does a finite well quantize too? Yes, but it supports only a limited number of bound states and their waves have external tails.

What does $V_0$ represent? It is the potential-energy scale separating bound behavior from escape or significant transmission.

Final synthesis

The $L$ determines which waves fit, and the allowed waves determine $E_n$. Quantization is not pasted onto the problem; it grows from confinement, boundary conditions, and wave nature.