Gaussian Wave Packets
From QM Translations, \(\langle x|p\rangle = N\exp\left[\frac{i}{\hbar}px\right]\).
\(\langle x'|x''\rangle = \delta(x'-x'') = \int dp'\langle x'|p'\rangle p'|x''\rangle = |N^2|\int dp' \exp\left[\frac{i}{\hbar}p'(x'-x'')\right]\)
From the Dirac Delta Function, \(\delta(x-x_0)=\frac{1}{2\pi}\int_\mathbb{R}\exp(i\kappa (x-x_0)) dk\).
So, \(\delta(x-x_0)=\frac{1}{2\pi\hbar}\int_\mathbb{R}\exp(i\hbar \kappa (x-x_0)) d(\hbar k)=\frac{1}{2\pi\hbar}\int_\mathbb{R}\exp(ip(x-x_0)) dp\).
Thus, \(\delta(x'-x'')=|N|^2\cdot 2\pi\hbar \delta (x'-x'') = |N^2|\int dp' \exp\left[\frac{i}{\hbar}p'(x'-x'')\right]\). Hence, \(N = \frac{1}{\sqrt{2\pi\hbar}}\).
Therefore, \(\langle x|p\rangle = \frac{1}{\sqrt{2\pi\hbar}}\exp(ipx/\hbar)\).
Relating Position Wavefunction to Momentum Wavefunction
\(\Psi_\alpha(x)=\langle x|\alpha\rangle = \int dp\langle x|p\rangle \langle p|\alpha\rangle = \int dp \frac{1}{\sqrt{2\pi\hbar}}\exp(ipx/\hbar)\Phi_\alpha(p)\).
This is just a Fourier, transform.
\(\Phi_\alpha(p) = \langle p|\alpha\rangle = \int dx \langle p|x\rangle\langle x|\alpha\rangle = \int dx \frac{1}{\sqrt{2\pi\hbar}}\exp(-ipx/\hbar)\Psi_\alpha(x)\).
Definition
\(\langle x|\alpha\rangle = \Psi_\alpha(x)=\frac{1}{\sqrt{d\sqrt{\pi}}|}\exp(-x^2/(2d^2))\)
Momentum Space
\(\Phi_\alpha(p)= \frac{1}{\sqrt{2\pi\hbar}}\int_\mathbb{R}\exp(-ipx/\hbar)\Psi_\alpha(x)dx = \frac{1}{\sqrt{2\pi\hbar}\sqrt{d\sqrt{\pi}}}\int_\mathbb{R}dx\exp(-ix/\hbar(p-\hbar k)-x^2/(2d)^2)\).
After completing the square we get,
\begin{align*} \Phi_\alpha(p) &= \frac{1}{\sqrt{2\pi\hbar}\sqrt{d\sqrt{\pi}}}\int_\mathbb{R}dx\exp(-ix/\hbar(p-\hbar k)-x^2/(2d)^2) \\ &= \frac{1}{\sqrt{2\pi\hbar}\sqrt{d\sqrt{\pi}}}\int_\mathbb{R}dx\exp -\left(\frac{x}{\sqrt{2}d}+\frac{i}{\hbar}(p-\hbar k)\frac{d}{\sqrt{2}}\right)^2 - \frac{-(p-\hbar k)^2 d^2}{2\hbar^2} \\ &= \frac{1}{\sqrt{2\pi\hbar}\sqrt{d\sqrt{\pi}}}\int dx \exp(-a^2)\exp(-(p-\hbar k)^2d^2/(2\hbar^2)) \\ &= \frac{1}{\sqrt{2\pi\hbar}\sqrt{d\sqrt{\pi}}} \int da \exp(-a^2)\exp(-(p-\hbar k)^2d^2/(2\hbar^2)) \\ &= \frac{1}{\sqrt{2\pi\hbar}\sqrt{d\sqrt{\pi}}} \sqrt{\pi}\exp(-(p-\hbar k)^2d^2/(2\hbar^2)) \\ &= \sqrt{\frac{d}{\hbar \sqrt{\pi}}}\exp\left[-\frac{(p-\hbar k)^2d^2}{2\hbar^2}\right] \\ &=\Phi_\alpha(p) \end{align*}The Gaussian Integral is, \(\int dx \exp(-\alpha x)=\sqrt{\frac{\pi}{\alpha}}\).
Properties
Probability
\(\mathcal{P}_{[x,x+dx]} = |\Psi_\alpha(x)|^2dx = \frac{1}{d\sqrt{\pi}}\exp(-x^2/d^2)dx\)
\(\mathcal{P}_{[p+p+dp]} = |\Phi_\alpha(p)|^2dp = \frac{d}{\hbar\sqrt{\pi}} \exp(-(p-\hbar k)^2d^2/(2\hbar^2))dp\).
Expectation Value
\(\langle X\rangle = 0 = \int_\mathbb{R}dx\psi^*_\alpha(x)x\psi_\alpha(x)\)
\(\langle P\rangle = \hbar k = \int_{\mathbb{R}^2} d^2x \Psi^*_\alpha(x'')(-i\hbar\frac{\partial}{\partial x})\delta(x'-x'')\Psi_\alpha(x)\)
Functional Properties
- Width at \(1/e\) times the height of the Gaussian is \(2d\).
Minimum Uncertainty
- \(\Psi_\alpha(x) = \frac{1}{\sqrt{d\sqrt{\pi}}}\exp(-x^2/(2d^2))\)
- \(\Phi_\alpha(x) = \sqrt{\frac{d}{\hbar\sqrt{\pi}}}\exp(-(p-\hbar k)^2d^2/(2\hbar^2))\)
- \(\langle X\rangle = 0\)
- \(\langle P\rangle = \hbar k\)
- \(\Delta a = \sqrt{\langle a^2\rangle - \langle a\rangle^2}\)
- \(\langle x^2\rangle = \frac{d^2}{2}\), \(\langle p^2\rangle = \frac{\hbar^2}{2d^2}+\hbar^2k^2\)
- \(\Delta x = \frac{d}{\sqrt{2}}\)
- \(\Delta p = \frac{\hbar}{d\sqrt{2}}\)
- \(\Delta x \Delta p = \frac{\hbar}{2}\)