What Does \mathbf{q_\perp} Represent in e^{i \mathbf{q_\perp \cdot x}}?

[MadMax]

If I have $$e^{i \mathbf{q_\perp \cdot x}}$$ what does it mean?Specifically what does the $$\mathbf{q_\perp}$$ mean?

thanks

 

[Dick]

It usually means the perpendicular component of q relative to something.

 

[MadMax]

Yes that's what I thought. Relative to what though?

Surely it can't mean relative to x because the dot product would imply that the term always = 0 right?

 

[Dick]

How could I guess 'relative to what'? I'd agree it's probably not x.

 

[MadMax]

The equation I'm dealing with which contains this term is

$$\epsilon(\mathbf{r})=\frac{i}{q_z} \int d^2 \mathbf{x} e^{i \mathbf{q_\bot \cdot x}}[\epsilon_2 e^{iq_z[H+h_2(\mathbf{x})]} - \epsilon_1 e^{iq_z h_1(\mathbf{x})}]$$

I guess it could be perpendicular to r... but what difference would that make? What would it mean?

 

[dextercioby]

It's probably perpendicular to the magnetic field intensity vector H.

 

[MadMax]

ahh see the H actually stands for height in this equation. :P h_x is a length also and purely a function of x.

But hmm perpendicular to H you say... that actually makes a lot more sense to me than any of the other variables if H were a vector, unfortunately its a mean separation, so that couldn't be it could it? I mean H is measured in a particular direction but... can you use that perpendicular symbol relative to something that's not a vector but measured in a particular dimension?

Cheers

 

[MadMax]

well thinking about it this is a 2 D problem using a radial or cartesian coordinate system. The radial dimensions are expressed by r and the cartesian dimensions are expressed by x= x_x + x_z.

Saying that we are dealing with something perpendicular to r makes no sense to me in the context of the system to be honest. Since it has cartesian symmetry but no radial symmetry. Although I could be missing somthing since the equation comes from a Fourier tranformation which I don't actually understand...

(a Fourier transform of the system

$$\epsilon(i f, r) = \epsilon_2(i f)$$ when $$H + h_2(x) \leq z < + \infty$$
$$\epsilon(i f, r) = 0$$ when $$h_1(x) < z < H + h_2(x)$$
$$\epsilon(i f, r) = \epsilon_1(i f)$$ when $$- \infty < z \leq h_1(x)$$

)


Saying its perpendicular to x is pointless. So I'm inclined to believe its either perpendicular to x_x or x_z. But which I don't know... :/ Nah actually though I bet if I actually understood the Fourier transform I'd understand what that q is perpendicular to :/ Can anyone help please? :(