- #1
ognik
- 643
- 2
Homework Statement
Assume ## \phi(k_x ) = \sqrt2 {\pi}## for ## \bar{k}_x - \delta \le k_x \le \bar{k}_x + \delta##, and ##= 0## for all other values of ##k_x##. Calculate ##\psi(x, 0)##, and show that ## \Delta x \Delta k_x \approx 1 ## holds if ## \Delta x## is taken as the width at half maximum.
Homework Equations
## \psi (x,0) = \frac{1}{\sqrt{2 \pi}} \int \phi (k_x) e^{i k_x x} dk_x ##
The Attempt at a Solution
## \psi (x,0) = \frac{1}{\sqrt{2 \pi}} \int_{\bar{k}_x - \delta } ^{\bar{k}_x + \delta} \sqrt{2 \pi} e^{i k_x x} dk_x ##
## = \frac{ e^{i k_x x}}{ix} |_{\bar{k}_x - \delta } ^{\bar{k}_x + \delta} ##
## = \frac{ e^{i \bar{k}_x x}}{x} 2 sin(x\delta) ##. If ##\delta## is small we can approximate to ## e^{i \bar{k}_x x} 2 \delta ##
Is this right please?
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How do I argue that ## \Delta x \Delta k_x \approx 1 ## holds if ## \Delta x## is taken as the width at half maximum?
I think that ##\psi_{max} = 2 \delta ## ?