- #1
ognik
- 643
- 2
Homework Statement
## \phi(k_x) = \begin{cases}\phantom{-}
\sqrt{2 \pi},\; \bar{k_x} - \frac{\delta}{2} \le k_x \le \bar{k_x} + \frac{\delta}{2} \\
- \sqrt{2 \pi},\; \bar{k_x} - \delta \le k_x \le \bar{k_x} - \frac{\delta}{2} \:AND \: \bar{k_x} + \frac{\delta}{2} \le k_x \le \bar{k_x} + \delta
\end{cases} ##
Calculate ## \psi(x, 0) ## and find ##\Delta x \Delta k_x ##
Homework Equations
## \psi(x, 0) = \frac{1}{\sqrt{2 \pi}} \int^{\infty}_{-\infty} \phi(k_x) e^{i k_x x} dk_x ##
The Attempt at a Solution
Firstly, ## \Delta k_x = 2 \delta ##
## \psi(x, 0) = -\int^{k_x - \frac{\delta}{2}}_{k_x - \delta} e^{i k_x x} dk_x +\int^{k_x + \frac{\delta}{2}}_{k_x - \frac{\delta}{2}} e^{i k_x x} dk_x -\int^{k_x + \delta}_{k_x + \frac{\delta}{2}} e^{i k_x x} dk_x ##
## = \frac{1}{i x} [ -e^{i k_x x (\bar{k_x} - \frac{\delta}{2})} + e^{i k_x x (\bar{k_x} - \delta)}
+ e^{i k_x x (\bar{k_x} +\frac{\delta}{2})} - e^{i k_x x (\bar{k_x} - \frac{\delta}{2})}
- e^{i k_x x (\bar{k_x} +\delta)} + e^{i k_x x (\bar{k_x} + \frac{\delta}{2})}
] ##
## = \frac{1}{i x} e^{i x \bar{k_x}} [ -2 (e^{i x \frac{\delta}{2}} - e^{-i x \frac{\delta}{2}}) - (e^{i x \delta} - e^{-i x \delta}) ##
## = \frac{1}{i x} e^{i x \bar{k_x}} [4 sin(x \frac{\delta}{2}) - 2sin(x \delta) ] ##
## = \frac{1}{i x} e^{i x \bar{k_x}} [ 2 \delta sinc (\frac{x \delta}{2}) - \delta sinc (x \delta )] ##
Taylor to 2 terms...
## [...] = 2 \delta (1 - \frac{(x \delta)^2}{6} ) - \delta (1 - \frac{(x \delta)^2}{6} ) ##
## = 2 \delta - \frac{x^2 \delta^3}{12} - \delta + \frac{x^2 \delta^3}{6} ##
## = \delta (1 + \frac{x^2 \delta^2}{12} ) ##
## \therefore \frac{1}{2} |\psi |^2 = \frac{\delta^2}{2} (1 + \frac{x^2 \delta^2}{12} ) ^2 ##
I hope this is right so far, please check? When I plot this in Wolfram I get a positive parabola, so no maximum, so I must have done something wrong but can't find it ...