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:

  1. \(\hbar\) (hbar) is hbar.

Notes:

  1. 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.

  2. 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:

  1. Wikipedia.

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

particle_momentum

momentum of the particle.

Symbol:

p

Latex:

\(p\)

Dimension:

momentum

particle_energy

energy of the particle.

Symbol:

E

Latex:

\(E\)

Dimension:

energy

position

position of the particle.

Symbol:

x

Latex:

\(x\)

Dimension:

length

time

time.

Symbol:

t

Latex:

\(t\)

Dimension:

time

law

psi = exp(I / hbar * (p * x - E * t))

Latex:
\[\psi = \exp{\left(\frac{i}{\hbar} \left(p x - E t\right) \right)}\]