Classical uncertainty principles

[danoonez]

Please check my work.

Using the first uncertainty principle:

$$\Delta x \Delta k \sim 1$$

derive the second uncertainty principle:

$$\Delta \omega \Delta t \sim 1$$


My work:

$$\Delta x \Delta k \sim 1$$

$$\frac {\Delta x} {\Delta t} = V \Rightarrow \Delta x = V \Delta t$$

$$V \Delta t \Delta k \sim 1$$

$$V_\textrm {group} = \frac {\Delta \omega} {\Delta k}$$

$$\frac {\Delta \omega} {\Delta k} \Delta t \Delta k \sim 1$$

$$\frac {\Delta k} {\Delta k} \Delta \omega \Delta t \sim 1$$

$$\Delta \omega \Delta t \sim 1$$

What do you think?

 

[Ed Quanta]

Looks good to me. Right on man. What were you uncertain about when you wrote this?

 

[danoonez]

Looks good to me. Right on man. What were you uncertain about when you wrote this?

I wasn't sure if my use of

$$V_\textrm {group} = \frac {\Delta \omega} {\Delta k}$$

was legal. But then I realized that the uncertainty principles are used for wave packets so it was fine.