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The group velocity for example of electron waves is given by the derivative of the dispersion relation: [itex]\frac{dE}{dp}=v[/itex] (this is for free electrons) [itex]^{1}[/itex]. Now the Heisenberg's uncertainty principle has two forms, one for position and momentum and the other for energy and time, namely:

[itex]\Delta x\Delta p \gtrsim h [/itex] [1] [itex]^{2}[/itex]

[itex]\Delta E\Delta t \gtrsim h [/itex] [2] [itex]^{2}[/itex]

(note these are approximate relations)

Dividing [2] nd expression from the [1] st gives:

[itex]\frac{\Delta E}{\Delta p}\frac{\Delta t}{\Delta x} \gtrsim 1 [/itex]

By letting the limits of the time and position changes tend to infinitesimally small values i.e. [itex]t\rightarrow 0 [/itex] and [itex]x\rightarrow 0 [/itex] we get the differential form:

[itex]\frac{dE}{dp}\frac{dt}{dx} \gtrsim 1 [/itex]

but [itex]\frac{dx}{dt} = v [/itex]

Hence

[itex]\frac{dE}{dp} \gtrsim v [/itex] which is the dispersion relation for a free particle.

I realise this may not be the most rigorous derivation, and it may be just a crackpottery as a result of my daydreaming. However, if there is any significance to this, I would like to hear an explanation to it. I could not find anything remotely close to this on the quantum book that I have which is by Alastair I. M. Rae.

Any input would be greatly appreciated,

Kind regards,

Cygni.

References:

(1) Experimental Physics, Modern Methods by R.A. Dunlap page 15.

(2) http://en.wikipedia.org/wiki/Uncertainty_principle

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# Dispersion relation and group velocity

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