Dispersion relation and group velocity

In summary, the group velocity for electron waves is given by the derivative of the dispersion relation. Heisenberg's uncertainty principle has two forms - one for position and momentum and the other for energy and time. By dividing the two forms, we get a differential form which can be used to derive the dispersion relation for a free particle. However, this method may not be entirely accurate and should be further investigated.
  • #1
Cygni
38
0
Hello there PF readers,

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 realize 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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  • #2
You can't simply divide those two relations, the result can also be a false statement, for example:

2>1

and

1.5>1.

If you divide the second by the first, you end up at

0.75>1,

which is obviously false.
 
  • #3
Thanks for your reply, Polyrhythmic,

I understand the reasoning behind the example you have showed, which is fine, it proves easily that I'm wrong. But could you elaborate more on why one can't simply divide those two relations?

Thanks in advance,

Kind regards,

Cygni.
 
  • #4
Well, when you divide two equations, you divide on both sides by the same quantity, it therefore remains a true statement. When you do that with inequalities, you're dividing both sides by different quantities, altering the original relation, leading to possibly wrong statements.
 

1. What is a dispersion relation?

A dispersion relation is a mathematical equation that describes the relationship between the frequency and wavelength of a wave. It is used to understand how a wave will behave in different mediums.

2. How is group velocity different from phase velocity?

Group velocity is the speed at which the overall shape of a wave travels, while phase velocity is the speed at which the individual crests and troughs of a wave move. In some cases, they can be the same, but in dispersive mediums, they can differ.

3. What is the significance of group velocity in wave propagation?

Group velocity is important because it determines how quickly a signal can be transmitted through a medium. In optical fibers, for example, the group velocity determines the speed of data transmission.

4. How does the dispersion relation affect the group velocity?

The dispersion relation plays a crucial role in determining the group velocity of a wave. In dispersive mediums, the group velocity is dependent on the frequency of the wave, meaning that waves of different frequencies will have different group velocities.

5. Can the group velocity be greater than the speed of light?

No, the group velocity cannot exceed the speed of light in a vacuum. This is a fundamental law of physics known as the universal speed limit. However, in certain mediums, the group velocity can be slower than the speed of light.

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