Last Updated on May 14, 2025 by Sushanta Barman

How are energy levels calculated?

In quantum mechanics, the infinite potential well (also called the 1D particle-in-a-box model) is one of the simplest systems used to illustrate energy quantization. It represents a particle confined between two rigid walls, where the potential \(V(x)\) is zero inside the well and infinite at the boundaries.

Physical setup

• The particle is free to move within a box of length \(L\), but cannot exist outside it due to the infinite potential barrier.
• The wave function must vanish at the boundaries (\(x = 0\) and \(x = L\)).

Quantum solution

The stationary states of the particle are obtained by solving the time-independent Schrödinger equation:
\[-\frac{\hbar^2}{2m} \frac{d^2\psi(x)}{dx^2} = E \psi(x)\]

Subject to boundary conditions:
\[\psi(0) = 0, \quad \psi(L) = 0\]

This leads to sinusoidal solutions inside the well:
\[\psi_n(x) = \sqrt{\frac{2}{L}} \sin\left(\frac{n\pi x}{L}\right), \quad n = 1, 2, 3, \dots\]

And the corresponding quantized energy levels:
\[E_n = \frac{n^2 h^2}{8 m L^2}\]
where, \(E_n\) is the energy of the \(n\)-th quantum level, \(h\) is Planck’s constant (\(6.626 \times 10^{-34} \ \mathrm{J \cdot s}\)), \(m\) is the mass of the particle (which depends on the selected particle), \(L\) is the length of the potential well (converted from nanometers to meters), and \(n\) is the quantum number, which takes positive integer values (1, 2, 3, …).

The energy is calculated in joules and then converted to electron volts (eV) using:
\[1 \ \mathrm{eV} = 1.602 \times 10^{-19} \ \mathrm{J}\]

What does this calculator do?

• You select a particle (e.g., electron, proton, atom).
• You specify the length \(L\) of the potential well in nanometers.
• You choose a maximum quantum number \(n\).
• The calculator then computes all energy levels from \(n = 1\) to \(n = n_{\text{max}}\).
• The results are displayed in a table and plotted as \(E_n\) vs. \(n\).