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module_1_summary [2021/02/24 06:37] – created adminmodule_1_summary [2021/03/01 18:35] (current) – [Phase and Group Velocity] admin
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 ====== Wave Packets ====== ====== Wave Packets ======
 +
 +The wavefunction $\psi(x,t) = e^{i(kx-\omega t)}$ represents a free particle with definite momentum $p=\hbar k$ abd $E = \hbar \omega$.  However, it is spread out over all of space.
 +
 +The wavefunction $\psi(x) = \delta(x-x_0)$ is localized at $x_0$, but has infinite momentum uncertainty.
 +
 +We can get more realistic solutions by superposing different momentum states:
 +
 +\[\psi(x,t) = \frac{1}{\sqrt{2\pi\hbar}} \int_{-\infty}^{+\infty} \phi(p) e^{i(px-Et)/\hbar}\,\D p\]
 +
 +For the Gaussian wavepacker
 +
 +\[\psi_0(x) = \left ( \frac{1}{2\pi\sigma_2}\right )^{\frac{1}{4}} e^{-x^2/4\sigma^2},\]
 +we derived $\Delta x = \sigma$, $\Delta p = \hbar/2\sigma$, so these are minimum uncertainty states that saturate the uncertainty relation
 +\[\Delta x \Delta p \geq \frac{\hbar}{2}.\]
 +
 +They are well-localized in both position and momentum, as illustrated below.
 +{{ :gaussians.png?direct&1000 |}}
  
 ===== Phase and Group Velocity ===== ===== Phase and Group Velocity =====
 +
 +{{ :phaseandgroup.png?direct&600 |}}
 +
 +The relationship $\omega(k)$ between angular frequency and wave number is called the //**dispersion relation**//, and depends on the equation of motion.  If $\omega = c k$ for some constant $c$ then there is no dispersion and the wave packet moves at speed $c$ without changing its shape of spreading.
 +
 +Generally, the //**phase velocity**// is the velocity of the fast oscillations and is given by
 +\[v_p = \frac{\omega}{k}.\]
 +
 +The //**group velocity**// is the velocity of the wave packet envelope and is given by
 +\[v_g = \frac{\D \omega}{\D k}.\]
 +
 +When there is dispersion, i.e. $\omega \neq ck$, the group and phase velocities differ.  In quantum mechanics, for a Hamiltonian
 +\[H = \frac{p^2}{2m} + V,\]
 +with $V$ constant, we have
 +\begin{align*}
 +v_p & = \frac{p}{2m} + \frac{V}{p}, & v_g & = \frac{p}{m},
 +\end{align*}
 +so the group velocity corresponds to the velocity of a classical particle.
  
 ===== Spreading of Wave Packets ===== ===== Spreading of Wave Packets =====
 +
 +{{ ::gaussianspread.png?direct&800 |}}
 +
 +When there is dispersion, wave packets spread in time.  For a Gaussian wave packet with initial position uncertainty $\sigma_0$, the uncertainty at time $t$ grows to
 +\[\sigma_t = \sigma_0 \sqrt{1 + \frac{\hbar^2 t^2}{4m^2\sigma_0^4}}.\]
 +
 +
 +