QM of a free particle in 1D

A particle in free space $𝑉 (𝑥) = 0$ has two-fold degenerate non-normalizable¹ (and hence non-physical) stationary states

Ψ 𝑘 (𝑥, 𝑡) = 1 \sqrt 2 𝜋 𝑒 𝑖 𝑘 𝑥 𝑒 - 𝑖 ℏ 𝑘 2 𝑡 / 2 𝑚

where $𝑘 = \pm \sqrt 2 𝑚 𝐸 𝑘 / ℏ$ with sign indicating direction of propagation. Since energy exhibits no quantisation, a general solution has the form

Ψ (𝑥, 𝑡) = 1 \sqrt 2 𝜋 \int \infty - \infty 𝜙 (𝑘) 𝑒 𝑖 𝑘 𝑥 𝑒 - 𝑖 ℏ 𝑘 2 𝑡 / 2 𝑚 𝑑 𝑘

where $𝜙 (𝑘)$ is the distribution of $𝑘$ within a wave packet, which can be found for normalized $Ψ (𝑥, 0)$ via the Fourier transform

𝜙 (𝑘) = F {Ψ (𝑥, 0)} (𝑘) = 1 \sqrt 2 𝜋 \int \infty - \infty Ψ (𝑥, 0) 𝑒 - 𝑖 𝑘 𝑥 𝑑 𝑥

Proof

The TISE reads

- ℏ 2 𝑚 𝑑 𝑑 𝑥 2 𝜓 = 𝐸 𝜓

or equivalently

𝑑 2 𝑑 𝑥 2 𝜓 = - 𝑘 2 𝜓

which has solutions

𝜓 (𝑥) = ˜ 𝐴 𝑒 𝑖 𝑘 𝑥 + ˜ 𝐵 𝑒 - 𝑖 𝑘 𝑥

which we split into left- ( $𝑘 < 0$ ) and right-moving ( $𝑘 > 0$ ) waves. Since there are no boundary conditions on $𝐴$ and $𝐵$ , there is no quantization of $𝑘$ . Furthermore these states are non-normalizable

Properties

The velocity of a stationary state $⟨ ˆ 𝑣 ⟩ = 𝑣 𝜑 = \sqrt 𝐸 2 𝑚$ , whereas the group velocity $𝑣 𝑔 = \sqrt 2 𝐸 𝑚$ matches the classical velocity.²
The probability flux for $Ψ 𝑘 (𝑥, 𝑡)$ is $𝐽 𝑘 (𝑥, 𝑡) = ℏ 𝑘 2 𝑚$ .

Proof of 2.

Applying ^1D we have

𝐽 𝑘 (𝑥, 𝑡) = 𝑖 ℏ 2 𝑚 (Ψ 𝑘 𝜕 𝜕 𝑥 Ψ * 𝑘 - Ψ * 𝑘 𝜕 𝜕 𝑥 Ψ 𝑘) = 𝑖 ℏ 2 𝜋 𝑚 (- 𝑖 𝑘 - 𝑖 𝑘) = ℏ 𝑘 𝜋 𝑚

proving ^P2.

QM of a free particle in 1D

Properties

Footnotes