Boundary conditions.
\[
\psi(0)=0,
\qquad
\psi(L)=0.
\]
These conditions allow only standing-wave solutions. The normalized stationary-state wave functions are
Normalized infinite-well wave functions.
\[
\psi_n(x)=\sqrt{\frac{2}{L}}\sin\!\left(\frac{n\pi x}{L}\right),
\qquad n=1,2,3,\dots
\]
The factor \(\sqrt{2/L}\) ensures normalization, meaning that the total probability of finding the particle somewhere
in the box is 1:
\[
\int_0^L |\psi_n(x)|^2\,dx = 1.
\]
Since the sine function oscillates, the sign of \(\psi_n(x)\) changes across the well. However, measurements of
position depend on the probability density rather than the sign of the wave function itself. The probability density is
Probability density.
\[
|\psi_n(x)|^2=\frac{2}{L}\sin^2\!\left(\frac{n\pi x}{L}\right).
\]
This quantity is always nonnegative. It tells us where the particle is more or less likely to be detected. For each
stationary state, the density has \(n\) lobes, one inside each half-wave of the sine pattern.
Nodes and lobes
The wave function has zeros at the walls and also at certain points inside the box. The internal nodes are the zeros
that occur strictly between \(x=0\) and \(x=L\). Their number is
\[
n-1.
\]
For example:
\[
n=1 \Rightarrow 0 \text{ internal nodes},
\qquad
n=2 \Rightarrow 1 \text{ internal node},
\qquad
n=3 \Rightarrow 2 \text{ internal nodes}.
\]
The probability density, meanwhile, has \(n\) peaks or lobes. For \(n=3\), the wave function shows three half-wave
sections and the probability density shows three distinct maxima. This distinction is important: the state has three
lobes but only two internal nodes.
Peak positions
The density maxima occur where the sine-squared factor reaches 1. These positions are
Probability-density maxima.
\[
x_{p,j}=\frac{(2j-1)L}{2n},
\qquad
j=1,2,\dots,n.
\]
These are the most likely positions within each lobe of the density pattern. They divide the well into evenly spaced
regions centered between neighboring nodes.
Wavelength and wave number
The standing-wave form also makes the spatial wavelength and wave number easy to identify:
\[
k=\frac{n\pi}{L},
\qquad
\lambda=\frac{2L}{n}.
\]
As the quantum number increases, the wave oscillates more rapidly, the wavelength becomes smaller, and the number of
nodes increases.
Connection to energy
Although this visualizer focuses on the wave shape and probability density, the same quantum number also determines the
energy:
\[
E_n=\frac{n^2\pi^2\hbar^2}{2mL^2}.
\]
This means higher-\(n\) states are both more oscillatory and more energetic. The spatial structure of the wave function
and the quantization of the energy come from the same boundary conditions.
Physical meaning
The infinite well is an idealized model, but it captures fundamental quantum ideas: normalization, standing waves,
quantization, nodes, and probability interpretation. It is used to introduce quantum confinement and to build intuition
for more advanced systems such as finite wells, quantum dots, and time-dependent wave-packet superpositions. Even
though no real wall is truly infinite, the particle-in-a-box model remains one of the clearest demonstrations that
confinement produces discrete wave patterns and measurable probability structures.
| Concept |
Main relation |
Meaning |
| Boundary conditions |
\(\psi(0)=0,\ \psi(L)=0\) |
Wave function vanishes at the infinite walls |
| Wave function |
\(\psi_n(x)=\sqrt{2/L}\sin(n\pi x/L)\) |
Allowed normalized standing-wave state |
| Probability density |
\(|\psi_n(x)|^2=\frac{2}{L}\sin^2(n\pi x/L)\) |
Position-space measurement distribution |
| Internal nodes |
\(n-1\) |
Zero crossings inside the well, excluding the walls |
| Density peaks |
\(x_{p,j}=\frac{(2j-1)L}{2n}\) |
Most likely positions within each lobe |
| Wave number |
\(k=n\pi/L\) |
Spatial oscillation rate of the standing wave |