QM in 1D position-space

QM of a free particle in 1D

A particle in free space has two-fold degenerate non-normalizable1 (and hence non-physical) stationary states

where with sign indicating direction of propagation. Since energy exhibits no quantisation, a general solution has the form

where is the distribution of within a wave packet, which can be found for normalized via the Fourier transform

Proof

The TISE reads

or equivalently

which has solutions

which we split into left- ( ) and right-moving ( ) waves. Since there are no boundary conditions on and , there is no quantization of . Furthermore these states are non-normalizable

Properties

  1. The velocity of a stationary state , whereas the group velocity matches the classical velocity.2
  2. The probability flux for is .
Proof of 2.

Applying ^1D we have

proving ^P2.

See also

#state/tidy | #lang/en | #SemBr

Footnotes

  1. But Dirac orthonormal ↩

  2. 2018. Introduction to quantum mechanics, §2.4, pp. 58–59. ↩