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Particle in a box
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Energy levels are quantised.
E_n = n²h²/8mL²A particle confined to a region of width L has energy levels E_n = n²h²/(8mL²), with nodes in ψ. The lowest energy is non-zero.
Electron in 1 nm box: E₁ = 0.38 eV.
Energy grows as n²; spacing grows with n.
In more detail
A particle confined to a box has discrete energy levels, E_n = n²h²/(8mL²). The levels increase with the square of the quantum number, and the lowest state has nonzero energy.
Real-world uses
- Quantum wells in lasers and semiconductors behave like particle-in-a-box systems.
- It is a simple model for electrons in conjugated molecules.
- Physicists use the particle in a box as a model for electrons in nanoscale wires and quantum dots.
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Frequently asked questions
Why is there no zero energy state?
Because the wavefunction must vanish at the walls, which requires a minimum energy.
What happens for larger boxes?
The energy levels become closer together and approach the classical continuum.
Why are the energies quantised?
The particle's wavefunction must fit into the box with zero value at the walls, which allows only certain wavelengths.
Related topics in Quantum Mechanics
Schrödinger equation (time-independent) · Operators and observables · Commutators · Free particle · Finite potential well · Quantum tunnelling