Free particle plane wave solution¶
In the absence of a potential field (\(U = 0\)) the wave function can be represented in the form of a plane wave.
Notation:
\(\hbar\) (
hbar
) ishbar
.
Notes:
The energy and momentum of a quantum particle are related by the equation \(E = p^2 / (2 m)\) where \(m\) is the mass of the quantum particle.
The wave function must be normalized, i.e. the integral of the square of its absolute value over the whole range of the spatial variable must equal 1, but the integral of the complex expontential diverges if taken over the real line. This is not a problem, though, because the states described by such a wave function would never be infinite (they would not be defined over the whole real line). Moreover, the correct solution can be represented by a linear combination of planar waves, which can be made convergent.
Links:
- dimensionless_wave_function¶
Dimensionless
wave_function
describing the particle’s state. The dimension coefficient has been omitted since this wave function cannot be normalized as it is, see Notes above.- Symbol:
psi
- Latex:
\(\psi\)
- Dimension:
dimensionless
- law¶
psi = exp(I / hbar * (p * x - E * t))
- Latex:
- \[\psi = \exp{\left(\frac{i}{\hbar} \left(p x - E t\right) \right)}\]