Calculate the energy levels of a quantum particle confined in a 1D box using the particle-in-a-box model. Enter the quantum number (n), particle mass (m), and box length (L) to get the energy level (Eₙ) in electron volts, plus the corresponding energy in joules. Based on the Schrödinger equation formula Eₙ = n²h²/(8mL²).
Energy Level Eₙ
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Energy in Joules
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Ground State Energy (n=1)
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Eₙ / E₁ Ratio
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de Broglie Wavelength
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The particle-in-a-box (or infinite square well) is a fundamental quantum mechanics model where a particle is free to move inside a perfectly rigid, impenetrable box but cannot escape it. It's used to illustrate how confinement leads to quantized (discrete) energy levels, a key difference between classical and quantum physics.
The energy of a particle in a 1D box is given by Eₙ = n²h²/(8mL²), where n is the quantum number (1, 2, 3, ...), h is Planck's constant (6.626×10⁻³⁴ J·s), m is the mass of the particle, and L is the length of the box. Energy increases with the square of the quantum number.
A quantum number of n=0 would imply zero energy (E=0), meaning the particle is at rest inside the box. This violates the Heisenberg Uncertainty Principle — if the particle's position is confined to the box, its momentum (and hence kinetic energy) cannot be exactly zero. Therefore, n must be a positive integer starting from 1.
The ground state corresponds to n=1, the lowest possible energy level a confined particle can have. It is given by E₁ = h²/(8mL²). All higher energy levels are integer multiples of n² times this ground state energy, so E₂ = 4E₁, E₃ = 9E₁, and so on.
Energy is inversely proportional to the square of the box length (E ∝ 1/L²). A smaller box leads to much higher energy levels — this is why electrons confined in nanoscale structures (like quantum dots) have much higher energies than those in macroscopic systems. Doubling the box length reduces energy to one quarter.
Despite being a simplified model, it approximates real systems like electrons in conjugated molecules (e.g., polyenes), quantum dots for LED displays and solar cells, electrons in nanowires, and nuclear models. It provides a first approximation for understanding quantum confinement effects in nanotechnology.
The de Broglie wavelength λ = h/p = h/√(2mE) represents the matter-wave character of the particle. For a particle in a box at energy level n, the box length always accommodates exactly n/2 wavelengths (L = nλ/2). The wavelength decreases as the quantum number increases because the particle moves faster.
As the quantum number n increases, the particle's wavefunction must fit more half-wavelengths inside the box. Each additional node requires shorter wavelengths, and since kinetic energy is proportional to the square of momentum (E = p²/2m) and momentum is inversely proportional to wavelength, energy scales as n².