Particle in a Box Energy Calculator

Calculate quantized energy levels for a particle confined in an infinite potential well.

n=1 is ground state
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Why Confinement Creates Discrete Energy Levels

In the one-dimensional infinite-well model, a particle confined to a region of length L cannot have arbitrary energy. Its wavefunction must be zero at both walls, so only standing waves that fit an integer number of half-wavelengths inside the box are allowed. That boundary condition quantizes momentum and produces discrete energy levels rather than a continuous range.

The energy grows as n2 and decreases as 1/L2. This means higher levels spread farther apart as n increases, while making the box larger rapidly lowers every level. Energy is also inversely proportional to particle mass, so a proton in the same box has much smaller level spacing than an electron because the proton is far heavier.

En = n2h2/(8mL2),   n=1,2,3,...
SymbolMeaningWhy it appears / units
nQuantum numberPositive integer; n=1 is the lowest allowed state.
hPlanck constant6.626×10−34 J·s.
mParticle masskg; heavier particles have smaller energy spacing.
LBox lengthm; the energy scales with 1/L2.
EnAllowed energyJ or eV; 1eV=1.602×10−19J.

The nonzero n=1 energy is called zero-point energy. The particle cannot have n=0 because that would make the wavefunction zero everywhere, meaning there is no particle state at all. The infinite walls are an idealization, but the model captures the central quantum idea that strong spatial confinement increases energy spacing.

Worked Examples

Example 1: Electron in 1nm box, n=1
E₁=π²ℏ²/(2m×(1e-9)²)
Result: 0.376 eV ground state
Quantum confinement at nanoscale
Example 2: n=2 level in 1nm box
E₂=4×E₁=4×0.376
Result: 1.504 eV — 4× ground state energy
E_n = n²E₁ — quadratic spacing
Example 3: Doubling the box length
Electron, n=1, L changes from 1nm to 2nm → E scales as 1/L2
Result: E1≈0.0940 eV
Doubling L reduces the ground-state energy to one-fourth of its 1nm value.
Example 4: Proton in the same 1nm box
Proton, n=1, L=1nm → E1=h2/(8mpL2)
Result: E1≈2.05×10−4 eV
The much larger proton mass makes the level spacing about 1836 times smaller than for an electron in the same box.

Common Mistakes

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Using n=0 as the ground state

The allowed quantum numbers are n=1,2,3,... . The ground state is n=1 and has nonzero energy.

⚠️
Forgetting that box length is squared

Energy varies as 1/L2. Doubling the box length divides every energy level by four, not by two.

⚠️
Entering nanometers directly into an SI formula

Convert 1nm to 1×10−9m before using h in joule-seconds and mass in kilograms.

Frequently Asked Questions

Physical meaning?
Particle confined in box cannot have zero energy (zero-point energy). Wavelength must fit: L = nλ/2. Only discrete wavelengths → discrete momenta → discrete energies.
Applications?
Quantum dots (CdSe nanoparticles): bandgap tuned by size, producing size-dependent colors. LED displays, biomedical imaging. Electron in semiconductor well: basis of transistors and lasers.
Why is the ground-state energy not zero?
A zero-energy particle would have zero momentum and an infinitely long de Broglie wavelength, which cannot satisfy the boundary conditions of a finite box. The lowest allowed standing wave has half a wavelength fitting across the box, so n=1 has a finite momentum and finite zero-point energy.
Why do energy levels scale as n squared?
The boundary condition allows wavelengths λn=2L/n, so momentum p=h/λ is proportional to n. Nonrelativistic kinetic energy is p2/(2m), which makes energy proportional to n2. The quadratic dependence therefore follows from standing-wave quantization combined with classical kinetic-energy form.
What happens to the energy levels if the box gets smaller?
Every energy level rises because En∝1/L2. Halving the box length makes each level four times larger and also increases the spacing between adjacent levels. This is why nanoscale confinement can noticeably change electronic energies and optical properties in quantum wells and quantum dots.
Is the infinite potential well a realistic physical box?
It is an ideal model in which the walls are perfectly impenetrable and the potential outside is infinite. Real quantum wells have finite barriers, so wavefunctions can penetrate slightly into the surrounding material. Even so, the infinite-well model is valuable for understanding quantization and estimating confinement trends.

Formula Explorer connections

Interpretation: This relationship connects quantized energy, wavelength, probability, nuclear mass or radioactive change. Assumption: Use the correct particle, quantum state, nuclide and energy units. Idealized potentials, nonrelativistic motion or single decay channels may be assumed.

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