Particle in a Box - 2D and 3D Energy Levels Calculator
Find the quantized energy levels of a particle in a 2D or 3D box using E = (h²/8m)(nx²/Lx² + ny²/Ly² + nz²/Lz²).
🧊 What is the 2D and 3D Particle in a Box Calculator?
This particle in a box calculator finds the quantized energy of a particle confined to a rectangular (2D) or box-shaped (3D) region, and flags whether that state shares its energy with another combination of quantum numbers. Choose 2D or 3D, enter the box dimensions and quantum numbers, and read off the energy in electronvolts and joules.
Extending the 1D particle in a box to more dimensions is straightforward mathematically, the total energy is just the sum of independent 1D contributions along each axis, each with its own quantum number and box length. But the extra dimensions introduce a genuinely new phenomenon: degeneracy, where different quantum-number combinations can produce the exact same total energy.
Degeneracy only appears when the box has some symmetry. In a perfect cube, for example, the states (2,1,1), (1,2,1), and (1,1,2) all have identical energy, since permuting which axis carries the "2" doesn't change the sum of squares. Stretch the box so the sides are unequal, and that symmetry breaks, splitting the once-degenerate levels apart, a real effect seen in the electronic structure of anisotropic quantum dots and nanocrystals.
This calculator is useful for students moving beyond the 1D case into multi-dimensional quantum confinement, semiconductor quantum wells and dots, and the general concept of degenerate energy levels, since it handles the arithmetic and flags degeneracy automatically.
📐 Formula
📖 How to Use This Calculator
Steps
💡 Example Calculations
Example 1 - Electron in a 1 nm square, ground state
Example 2 - Same square, an excited (nx=2, ny=1) state
Example 3 - Electron in a 1 nm cube, ground state
❓ Frequently Asked Questions
🔗 Related Calculators
What is the 2D and 3D particle in a box model?
It extends the 1D infinite square well to two or three dimensions, a particle confined inside a rectangular (2D) or box-shaped (3D) region with impenetrable walls. The energy now depends on a set of quantum numbers, one for each dimension, and the total energy is the simple sum of the 1D energy contribution from each direction.
What is the formula for 3D particle-in-a-box energy?
E = (h² / 8m) × (nx²/Lx² + ny²/Ly² + nz²/Lz²), where nx, ny, nz are independent positive-integer quantum numbers for each dimension, and Lx, Ly, Lz are the box's side lengths. Dropping the nz²/Lz² term gives the 2D version for a rectangular sheet.
What is degeneracy in this context?
Degeneracy occurs when two or more different combinations of quantum numbers give exactly the same total energy. This only happens when the box has some symmetry, for example a perfect cube (Lx = Ly = Lz), where permuting the quantum numbers, such as (2,1,1), (1,2,1), and (1,1,2), leaves the energy unchanged, giving three degenerate states.
Why doesn't the 1D particle in a box have degenerate states?
In 1D there is only one quantum number, so each distinct value of n gives a distinct energy, there is nothing to permute. Degeneracy only appears once there are multiple independent quantum numbers that can be swapped while keeping the sum of their squares (weighted by the box dimensions) the same.
How does stretching the box affect degeneracy?
If the box is stretched so that Lx, Ly, and Lz are no longer equal, the symmetry that caused degenerate states is broken. States that used to share an energy in the cube now have slightly different energies, since each dimension contributes a different amount depending on its own length, an effect called degeneracy lifting or splitting.
What real systems use the 2D and 3D particle-in-a-box model?
2D quantum wells and quantum wires in semiconductor devices are modeled with the 2D version, while 3D quantum dots, a nanocrystal confining electrons in all three directions, use the full 3D formula. The model also gives useful qualitative insight into electrons trapped in more complex 3D potentials, such as nanoparticles or artificial atoms.
How do I find the ground state in 3D?
The ground state is always nx = ny = nz = 1, the lowest possible value for each quantum number, since the energy contribution from each dimension is minimized at n = 1. Just as in the 1D case, the particle can never have zero energy, even in the ground state, a consequence of the uncertainty principle applied along each axis.
Can a 3D box have more complex degeneracy than a 2D box?
Yes. A cube has richer degeneracy patterns than a square, because there are more ways to permute three quantum numbers than two, and additional 'accidental' degeneracies can occur where different sums of squares happen to coincide (for example nx²+ny²+nz² = 9 for both (2,2,1) and (3,0,0)-type combinations, when zero were allowed, illustrating why symmetry alone doesn't capture every degeneracy in higher dimensions).
Does this model apply to non-rectangular boxes?
No, this exact separable solution only works for a rectangular (2D) or box-shaped (3D) region with flat walls meeting at right angles. Other shapes, such as a circular or spherical confining region, require different mathematics (Bessel functions for a circle, spherical harmonics for a sphere), though the same core idea, that confinement quantizes the energy, still applies.
How many quantum numbers does a 3D box need?
Exactly three, one independent positive integer for each spatial dimension (nx, ny, nz), reflecting the three degrees of freedom a particle has in 3D space. This is a general pattern in quantum mechanics: the number of quantum numbers needed to specify a state matches the number of independent directions or degrees of freedom in the system.