Gaussian Wave Packets

From QM Translations, $⟨ x | p ⟩ = N \exp [\frac{i}{ℏ} p x]$ .

$⟨ x^{'} | x^{″} ⟩ = δ (x^{'} - x^{″}) = \int d p^{'} ⟨ x^{'} | p^{'} ⟩ p^{'} | x^{″} ⟩ = | N^{2} | \int d p^{'} \exp [\frac{i}{ℏ} p^{'} (x^{'} - x^{″})]$

From the Dirac Delta Function, $δ (x - x_{0}) = \frac{1}{2 π} \int_{R} \exp (i κ (x - x_{0})) d k$ .

So, $δ (x - x_{0}) = \frac{1}{2 π ℏ} \int_{R} \exp (i ℏ κ (x - x_{0})) d (ℏ k) = \frac{1}{2 π ℏ} \int_{R} \exp (i p (x - x_{0})) d p$ .

Thus, $δ (x^{'} - x^{″}) = | N |^{2} \cdot 2 π ℏ δ (x^{'} - x^{″}) = | N^{2} | \int d p^{'} \exp [\frac{i}{ℏ} p^{'} (x^{'} - x^{″})]$ . Hence, $N = \frac{1}{\sqrt{2 π ℏ}}$ .

Therefore, $⟨ x | p ⟩ = \frac{1}{\sqrt{2 π ℏ}} \exp (i p x / ℏ)$ .

Relating Position Wavefunction to Momentum Wavefunction

$Ψ_{α} (x) = ⟨ x | α ⟩ = \int d p ⟨ x | p ⟩ ⟨ p | α ⟩ = \int d p \frac{1}{\sqrt{2 π ℏ}} \exp (i p x / ℏ) Φ_{α} (p)$ .

This is just a Fourier, transform.

$Φ_{α} (p) = ⟨ p | α ⟩ = \int d x ⟨ p | x ⟩ ⟨ x | α ⟩ = \int d x \frac{1}{\sqrt{2 π ℏ}} \exp (- i p x / ℏ) Ψ_{α} (x)$ .

Definition

$⟨ x | α ⟩ = Ψ_{α} (x) = \frac{1}{\sqrt{d \sqrt{π}} |} \exp (- x^{2} / (2 d^{2}))$

Momentum Space

$Φ_{α} (p) = \frac{1}{\sqrt{2 π ℏ}} \int_{R} \exp (- i p x / ℏ) Ψ_{α} (x) d x = \frac{1}{\sqrt{2 π ℏ} \sqrt{d \sqrt{π}}} \int_{R} d x \exp (- i x / ℏ (p - ℏ k) - x^{2} / (2 d)^{2})$ .

After completing the square we get,

\begin{aligned} Φ_{α} (p) & = \frac{1}{\sqrt{2 π ℏ} \sqrt{d \sqrt{π}}} \int_{R} d x \exp (- i x / ℏ (p - ℏ k) - x^{2} / (2 d)^{2}) \\ = \frac{1}{\sqrt{2 π ℏ} \sqrt{d \sqrt{π}}} \int_{R} d x \exp - {(\frac{x}{\sqrt{2} d} + \frac{i}{ℏ} (p - ℏ k) \frac{d}{\sqrt{2}})}^{2} - \frac{- (p - ℏ k)^{2} d^{2}}{2 ℏ^{2}} \\ = \frac{1}{\sqrt{2 π ℏ} \sqrt{d \sqrt{π}}} \int d x \exp (- a^{2}) \exp (- (p - ℏ k)^{2} d^{2} / (2 ℏ^{2})) \\ = \frac{1}{\sqrt{2 π ℏ} \sqrt{d \sqrt{π}}} \int d a \exp (- a^{2}) \exp (- (p - ℏ k)^{2} d^{2} / (2 ℏ^{2})) \\ = \frac{1}{\sqrt{2 π ℏ} \sqrt{d \sqrt{π}}} \sqrt{π} \exp (- (p - ℏ k)^{2} d^{2} / (2 ℏ^{2})) \\ = \sqrt{\frac{d}{ℏ \sqrt{π}}} \exp [- \frac{(p - ℏ k)^{2} d^{2}}{2 ℏ^{2}}] \\ = Φ_{α} (p) \end{aligned}

The Gaussian Integral is, $\int d x \exp (- α x) = \sqrt{\frac{π}{α}}$ .

Properties

Probability

$P_{[x, x + d x]} = | Ψ_{α} (x) |^{2} d x = \frac{1}{d \sqrt{π}} \exp (- x^{2} / d^{2}) d x$

$P_{[p + p + d p]} = | Φ_{α} (p) |^{2} d p = \frac{d}{ℏ \sqrt{π}} \exp (- (p - ℏ k)^{2} d^{2} / (2 ℏ^{2})) d p$ .

Expectation Value

$⟨ X ⟩ = 0 = \int_{R} d x ψ_{α}^{*} (x) x ψ_{α} (x)$

$⟨ P ⟩ = ℏ k = \int_{R^{2}} d^{2} x Ψ_{α}^{*} (x^{″}) (- i ℏ \frac{\partial}{\partial x}) δ (x^{'} - x^{″}) Ψ_{α} (x)$

Functional Properties

Width at $1 / e$ times the height of the Gaussian is $2 d$ .

Minimum Uncertainty

$Ψ_{α} (x) = \frac{1}{\sqrt{d \sqrt{π}}} \exp (- x^{2} / (2 d^{2}))$
$Φ_{α} (x) = \sqrt{\frac{d}{ℏ \sqrt{π}}} \exp (- (p - ℏ k)^{2} d^{2} / (2 ℏ^{2}))$
$⟨ X ⟩ = 0$
$⟨ P ⟩ = ℏ k$
$Δ a = \sqrt{⟨ a^{2} ⟩ - ⟨ a ⟩^{2}}$
$⟨ x^{2} ⟩ = \frac{d^{2}}{2}$ , $⟨ p^{2} ⟩ = \frac{ℏ^{2}}{2 d^{2}} + ℏ^{2} k^{2}$
$Δ x = \frac{d}{\sqrt{2}}$
$Δ p = \frac{ℏ}{d \sqrt{2}}$
$Δ x Δ p = \frac{ℏ}{2}$