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

• Uncertainty principle for a free particle in 1D position-space

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Footnotes

1. But Dirac orthonormal ↩

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