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Free particle
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Plane waves with continuous energy.
With V = 0 the solutions are plane waves e^{ikx} with E = ħ²k²/2m, continuous energy. Localised wave packets spread with time.
A moving electron beam.
Plane waves aren't normalisable.
In more detail
A free particle has no potential and its wavefunction is a plane wave, with a continuous range of energies. Its momentum is well defined but its position is spread out.
Real-world uses
- Electron beams in vacuum behave as free particles over short distances.
- Scattering theory considers particles approaching from far away as free.
- Physicists use free-particle plane waves to describe electrons moving through a uniform region of a material.
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Frequently asked questions
Is a free particle normalisable?
Not strictly, so physicists use wave packets to describe localised free particles.
Why is momentum well defined?
Because a plane wave has a single wavelength and therefore a single momentum.
Why are plane waves not normalisable?
They extend infinitely in space, so they describe idealised states rather than localised particles.
Related topics in Quantum Mechanics
Operators and observables · Commutators · Particle in a box · Finite potential well · Quantum tunnelling · Tunnelling applications