Reduced Compton Wavelength appearing in different equations

[Mystery111]

Ok, so my question is an odd one, but it's something that I've wondered for a while now. Is there any physical meaning why the reduced compton wavelength appears as

 

[math]\frac{\hbar}{Mc^2}[/math]

 

in certain equations, for example, like the fine structure constant, but then it appears as the reciprocal in other equations

 

[math]\frac{Mc^2}{\hbar}[/math]

 

such as the Klein Gorden equation or even the Dirac Equation...?

Edited October 18, 2011 by Mystery111

[ajb]

The Compton and reduced Compton wavelength only differ by a factor of [math]2 \pi[/math], so for "back of the envelope calculations" they are the same. By this I mean that we can use either to give us a length scale on which quantum field theory should really come into play.

 

For the electron this scale defines the point at which quantum electrodynamics is required. We have

 

[math]\lambda = 2.4 \times 10^{-12}\: m[/math] so dividing by the [math]2 \pi[/math] gives

 

[math]0.4 \times 10^{-12}\: m[/math].

 

As a length scale these are essentially the same, anywhere near [math]10^{-12}\: m[/math] and quantum electrodynamics should be the theory to use. For sure on any smaller scales quantum field theory will be essential.

[Mystery111]

Hi ajb, are you telling me it doesn't matter whether you write it as: hbar /Mc^2 and Mc^2/hbar... they produce the same results?

 

I don't know if you understood my question... I wasn't exactly asking how much the Compton and reduced Compton wavelength varied. I meant the way the reduced Compton wavelength makes appearances in different equations in different forms.

 

Thanks!

[ajb]

The reduced Compton wavelength is usually understood as defining a kind of cut-off point for quantum field theory. It gives a natural length scale to quantum field theory. As the reduced Compton wavelength and the full Compton wavelength are really of the same kind of order I doubt anyone would argue which one you wanted to use to define this scale.

 

But the reduced one seems to be the truly correct thing to use as it comes up naturally in the equations.

[Mystery111]

Right ... I understand that sure. But.... .... lol.. ... the RCW appears originally as

 

[math]\frac{\lambda}{2\pi} = \frac{\hbar}{Mc^2}[/math]

 

In some equations, this expression is flipped

 

[math]\frac{Mc^2}{\hbar}[/math]

 

Why is this?

 

 

Thanks

[ajb]

I doubt there is any deep meaning to this, other than on dimensional grounds.

Edited October 18, 2011 by ajb

[Mystery111]

Ok, I suspected that. I also thought another reason might be due to how the equations are derived, but thank you ajb!!!!

[ajb]

I also thought another reason might be due to how the equations are derived...

 

These are more or less the same thing. For example "one over" the Compton wavelength features in the Rydberg constant, which you can get at from quantum mechanics.

 

The Rydberg constant determines the change in energy or equivalently "one over" the wavelength for atomic transitions of the hydrogen atom. Because of this on dimensional grounds we want "one over" a length, and thus the Compton wavelength (reduced or not) should appear as "one over".

 

I don't think there is anything deep in this.

Edited October 18, 2011 by ajb

[Mystery111]

Hmmm nice. Made me think of it another way.

 

Thanks!