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Bandwidth Theorem: Find Min Angular Freq in Propagating Wavepacket
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[QUOTE="collinsmark, post: 5081936, member: 114325"] Hello bananabandana, I must ask, did you get the bandwidth theorem, [itex] \Delta f \Delta t \gt 1 [/itex] straight from you textbook/coursework, or from some other source? I ask because I've also seen this uncertainty principle (called the "bandwidth theorem" here) expressed as [tex] \Delta \omega \Delta t \approx 1 [/tex] elsewhere. Notice that the [itex] 2 \pi [/itex] is folded into it already, the way I've seen it. Admittedly this definition is an ill-defined version with the "delta" (what is specifically meant by "delta"?) and the approximation symbol. You can make it more precise by working with standard deviations (instead of deltas), and an inequality [and there might be a factor of 2 that fits into that one], but I'm getting off topic. Whatever the case, the above way is a way I have seen it before. If you use the [itex] \Delta \omega \Delta t \approx 1 [/itex] version, you can get the given answer after making some appropriate substitutions. First, you might try to work toward an equivalent form of the relationship in terms of [itex] \Delta x [/itex] and [itex] \Delta k [/itex]. You can do this in part by noting that velocity [itex] v = \frac{\Delta x}{\Delta t} [/itex], along with some other substitutions. Once you get there, you can invoke your [itex] v_g = \frac{\Delta \omega}{\Delta k} [/itex] as a near final step. ([itex] \Delta x [/itex] is the same thing as [itex] L_0 [/itex] by the way.) [/QUOTE]
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Bandwidth Theorem: Find Min Angular Freq in Propagating Wavepacket
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