Quantum Well — Energy Levels

Calculate energy levels and visualize wave functions

Parameters
Energy Diagram
Wave Function ψ(x)
Infinite Quantum Well

In an infinite well, the electron is completely localized inside the potential barrier. The energy spectrum is discrete and calculated by an analytical formula:

$$ E_n = \frac{n^2 \pi^2 \hbar^2}{2mL^2} \quad (n = 1, 2, 3, \ldots) $$

where: $n$ — quantum number, $\hbar$ — reduced Planck constant, $m$ — electron mass, $L$ — well width

$$ \psi_n(x) = \sqrt{\frac{2}{L}} \sin\!\left(\frac{n\pi x}{L}\right) $$
Finite Quantum Well

In a finite well, the potential barrier is finite (V₀). The Schrödinger equation is solved numerically (brentq method):

$$ -\frac{\hbar^2}{2m} \frac{d^2\psi}{dx^2} + V(x)\psi = E\psi $$

A finite well contains only a limited number of bound states. The electron "leaks" outside the barrier — this is the quantum tunneling effect.

Real Systems
System L (nm) V₀ (eV) E₁ (eV)
GaAs კვანტური ჭა100.3~0.006
Si/SiO₂ ჭა53.1~0.09
InGaAs/InP ჭა80.5~0.015
მარტივი მოდელი1∞~0.376
Step-by-Step Guide
1
Choose Well Type
Infinite well — simple model with analytical solution. Finite well — realistic, solved with SciPy root-finding.
2
Enter Width
Enter L in nanometers. Real quantum wells: 1–20 nm. Smaller L → higher energies.
3
Calculate
The Calculate button runs Python/SciPy on the VPS. Results are returned in eV.
4
Energy Diagram
On the right you see the well with energy levels. Plotly interactive features (zoom, hover) are available.
5
Wave Function
Use n= buttons to select the quantum number. The graph shows ψₙ(x) and |ψₙ|² (probability density).
Interpreting Results

E₁ is the ground state energy (zero-point energy). Energy levels scale as n² in an infinite well. Doubling L → E₁ decreases 4×. A finite well contains fewer levels than an infinite one — depends on barrier height.

Self-Assessment

Q1. In an infinite quantum well, the width L doubles. What happens to E₁?

A Doubles
B Decreases 4×
C Unchanged
D Halves

Q2. Which parameters determine the energy levels of a quantum well?

A Well width only
B Particle mass only
C Well width, particle mass, and quantum number
D Quantum number only

Q3. The energy difference between n=1 and n=2 states is how many times E₁?

A 3E₁
B 2E₁
C E₁
D 4E₁