Quantization

Once a quantum system is described by a wavefunction, the next question is: which states are actually allowed? The time-independent Schrodinger equation does not accept every imaginable shape. Boundary conditions, normalizability, and the form of the potential select only certain stationary states. That is what quantization means: a bound system admits a discrete set of energies rather than a continuous range.

Time-Independent Schrodinger Equation

For a particle in one dimension, the time-independent Schrodinger equation is

This equation plays a role somewhat analogous to Newton’s second law in classical mechanics: it connects the system’s environment, represented by the potential energy function $V(x)$, to the allowed states $\psi$ and their energies $E$.

Rigid Box

Inside a rigid one-dimensional box (aka infinite potential well), the walls force the wavefunction to vanish at the boundaries:

Those conditions mean only standing waves that fit exactly inside the box are allowed. The resulting stationary states are

with energies

The allowed energies are therefore quantized, not continuous. There is no $n=0$ bound state in this model, so even the lowest state carries nonzero energy.

Quantization Beyond Hard Walls

The rigid box is the cleanest example, but discrete energies are not unique to hard-wall boundaries.

For the harmonic oscillator, the potential is smooth rather than abrupt:

Here the wavefunction must stay normalizable as $|x| \to \infty$. That requirement picks out the equally spaced energy ladder

The ground state already has the nonzero zero-point energy $\frac{1}{2}\hbar\omega$, and each excited state adds another node.

For the hydrogen atom, the potential comes from the Coulomb attraction between the proton and electron.

As a more complex system, we can expect the wavefunction to require more than one integer $n$ to describe the allowed states. The hydrogen atom states are labeled with three quantum numbers $n$, $l$, and $m$. We do not need the full derivation here to see the main quantization pattern:

The radial part of the hydrogen wavefunction depends on both $n$ and $l$, while the angular part is carried by spherical harmonics. In the explorer below, the hydrogen panel focuses on that radial component so you can compare its nodes and radial probability pattern with the box and oscillator states.

As you explore the diagram below, try this sequence:

1. In infinite-well mode, raise the quantum number and count the added nodes.
2. Switch to the harmonic oscillator and compare the equal energy spacing with the box’s $n^2$ growth.
3. In the hydrogen panel, keep $n$ fixed and change $l$ to see the radial shape change even when the simple energy formula still depends only on $n$.
4. Compare the probability view in all three systems and look for how nodes in the amplitude create dips in the probability density.

Note that it’s convention to overlay the wavefunction $\psi(x)$ on the same plot describing the allowed energies of a system. Below, the potential energy function is also drawn in black. It’s a lot at first, but helpful once you start to see the connection between things like frequency and energy.