3.7605e-1
eV
1.5042e+0
eV
1.1281e+0
eV
1.0990e+3
nm
2.7278e+14
Hz
3.7605e-1
eV
1.5042e+0
eV
1.1281e+0
eV
1.0990e+3
nm
2.7278e+14
Hz
The particle in a box (infinite square well) is the most fundamental exactly solvable problem in quantum mechanics and serves as the gateway to understanding energy quantization, wavefunctions, and quantum transitions. This Particle in a Box Calculator computes the energy levels for any two quantum states and determines the transition energy, wavelength, and frequency of the photon emitted or absorbed during a quantum transition. The model describes a particle constrained to move freely within a rigid box of length L, with impenetrable walls. Despite its simplicity, this model accurately predicts the absorption spectra of conjugated organic molecules, the optical properties of quantum dots and quantum wires, and provides qualitative insight into semiconductor band gaps. The calculator supports any particle mass, making it applicable to electrons, protons, neutrons, and other quantum particles in confined geometries.
The allowed energy levels for a particle of mass m in a one-dimensional box of length L are:
$$E_n = \frac{n^2 h^2}{8mL^2}$$
where n = 1, 2, 3, ... is the quantum number and h is Planck's constant. The transition energy between levels n₁ and n₂ is:
$$\Delta E = E_{n_2} - E_{n_1} = \frac{h^2}{8mL^2}(n_2^2 - n_1^2)$$
The corresponding photon wavelength and frequency are:
$$\lambda = \frac{hc}{\Delta E} = \frac{8mL^2 c}{h(n_2^2 - n_1^2)}$$
$$\nu = \frac{\Delta E}{h} = \frac{h(n_2^2 - n_1^2)}{8mL^2}$$
Selection rules: in the basic model, any transition between levels is allowed. In practice for electromagnetic transitions, Δn = ±1 transitions are strongest for symmetric perturbations. The wavefunctions are standing waves: ψ_n(x) = √(2/L) sin(nπx/L), with (n−1) nodes.
The transition energy determines whether the photon falls in the UV, visible, or infrared region of the electromagnetic spectrum. For electrons in molecular-sized boxes (0.5–2 nm), transitions typically produce UV-visible photons, explaining why conjugated molecules are colored. Larger boxes give smaller transition energies (longer wavelengths), which is why longer conjugated chains absorb at longer wavelengths. For quantum dots (2–10 nm), tuning the particle size controls the emission color—smaller dots emit blue/UV light, larger dots emit red. The energy spacing between adjacent levels (n and n+1) increases with n: ΔE = h²(2n+1)/(8mL²), meaning higher transitions require more energy. If the transition wavelength is negative, it means n₂ < n₁ (emission rather than absorption).
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The 1→2 transition in a 1 nm box requires 1.13 eV, producing a near-infrared photon at ~1099 nm. This is the lowest-energy (HOMO→LUMO) transition.
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Modeling a conjugated dye with 7 double bonds (L ≈ 1.4 nm), the 3→4 transition (HOMO→LUMO for 6 π-electrons) gives ~1156 nm. Real dyes absorb at shorter wavelengths due to finite potential well effects.
It describes a quantum particle confined between two impenetrable walls separated by distance L. Inside the box, the particle moves freely (V = 0); at the walls, the potential is infinite. The solutions are standing waves with quantized energies E_n = n²h²/(8mL²).
In conjugated organic molecules, π-electrons are delocalized along the molecular chain. The chain length serves as L, and the lowest unoccupied → highest occupied transition predicts the absorption wavelength. This free electron model successfully explains color trends in polyene dyes.
The n² dependence comes from the boundary conditions requiring an integer number of half-wavelengths to fit in the box: L = nλ/2. Since E = p²/(2m) and p = h/λ = nh/(2L), energy goes as n². This means energy level spacing increases with n.
Transition wavelength depends on ΔE = h²(n₂² − n₁²)/(8mL²). It increases (longer wavelength, redder) with: larger box L, heavier particle mass, and smaller difference in n² values. It decreases (shorter wavelength, bluer) with smaller boxes.
Quantum dots are semiconductor nanocrystals that confine electrons in all three dimensions. Their band gap energy depends on particle size through the E ∝ 1/L² relationship. By controlling dot size during synthesis (2–10 nm), the emission color is tuned from UV through visible to infrared.
The ground state (n = 1) energy E₁ = h²/(8mL²) is always greater than zero. This zero-point energy is a purely quantum effect—classically, a particle can have zero kinetic energy. It represents the minimum energy required by the uncertainty principle.
As L → ∞, energy levels become infinitely closely spaced (E_n → 0), approaching a continuous spectrum. This is the correspondence principle—quantum mechanics reduces to classical mechanics for large systems.
Yes. For a 3D rectangular box: E(n_x,n_y,n_z) = h²/(8m)(n_x²/L_x² + n_y²/L_y² + n_z²/L_z²). Each dimension contributes independently, and degeneracy occurs when different quantum number combinations give the same energy.
Degeneracy occurs when multiple distinct quantum states have the same energy. In a 2D or 3D square box, states like (1,2) and (2,1) have equal energy. Degeneracy is broken if the box dimensions differ (rectangular box).
It provides hands-on experience with quantum concepts: energy quantization, spectroscopic transitions, and the effect of confinement on electronic properties. Students can explore how molecular size affects absorption spectra and verify calculations from physical chemistry coursework.
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