Editing Wave packet (section)

=== The 2D case ===
[[File:2D Gaussian Quantum Wave Packet.gif|250px|thumb|A 2D gaussian quantum wave packet. The color (yellow green blue) indicates the phase of the wave function <math>\psi</math>, its brightness indicates <math>| \psi | ^2 / | \psi |_{max} ^2</math>. <math>k_{0x}= k_0</math>, <math>k_{0y}=0</math>]]

A gaussian 2D quantum wave function:

<math>\psi(x,y,t)=\psi(x,t) \psi(y,t)</math>
 
<math> \psi (x,t) = \left(\frac{2a ^2}{\pi}\right) ^{1/4} \frac{e ^{i \phi}}{ \left( a ^4 + \frac{4 \hbar ^2 t ^2}{m ^2} \right) ^{1/4}} e ^{i k_0 x} \exp\left[ - \frac{\left(x - \frac{\hbar k_0}{m} t\right) ^2 }{a^2 + \frac{2 i \hbar t}{m}} \right]</math>

where{{sfn|Cohen-Tannoudji|Diu|Laloë|2019|p=59}}

<math> \phi = - \theta - \frac{\hbar k_0 ^2}{2 m} t</math>

<math>\tan (2 \theta) = \frac{2 \hbar t}{m a ^2}</math>