Understanding the Wavevector: Exploring Its Role in Optics and K-Space

[iLIKEstuff]

What is wavevector?

So I've been studying optics on my own for about 2 months now and I am having a lot of trouble understanding what exactly is the wavevector

$$\overrightarrow{k} = (k_x , k_y , k_z)$$

where wavenumber k is

$$k = | \overrightarrow{k} | = \sqrt{{k_x}^2 + {k_y}^2 + {k_z}^2}$$

i was told when i first started that the meaning of wave vector would just come to me as i digged deeper into optics, but now I'm looking at E(k) vs k diagrams for semiconductors and this talk about k-space is confusing the heck out of me.

so i know $$k= \frac{2 \pi}{ \lambda}$$

and that the wave number k is the coefficient in the helmholtz equation

$${\nabla}^{2} U + {k}^{2} U = 0$$ where U is the complex amplitude and a function of x,y,z,t

but how does this relate to the wave vector? i know it doesn't expand like
$$\frac{{\partial}^{2}}{{\partial x}^{2}} \left ( U (x,t) \right ) + {k_x}^{2} U(x,t) =0$$
$$\frac{{\partial}^{2}}{{\partial y}^{2}} \left ( U (y,t) \right ) + {k_y}^{2} U(y,t) =0$$
$$\frac{{\partial}^{2}}{{\partial z}^{2}} \left ( U (z,t) \right ) + {k_z}^{2} U(z,t) =0$$

because k is the wave number. I'm lost.

perhaps i am approaching this concept in the wrong way. i would gladly appreciate any pointers on how to understand this menacing construct

i'm looking for a way to understand the wave vector in the context of understanding k-space and E(k) vs k diagrams to then understand optical absorption by transitions across band gaps for different materials. but also i simply just don't understand why they break up the wavevector into components whose magnitude is the wave number which is clearly (and understandably) defined as inversely proportional to the wavelength

thanks guys.

 

[Redbelly98]

k is just a convenient mathematical way of expression both the direction of a wave and it's wavelength. (lambda = 2 pi / |k|, as you know)

 

[DeepSeeded]

i simply just don't understand why they break up the wavevector into components

If it is a vector then it has components, U looks like a 4D vector to me, and if you want the second gradiant of a vector your going to need to break the equation into it's components so that you can take the second partial derivative of each component and then "put it back together".

 

[Cthugha]

i was told when i first started that the meaning of wave vector would just come to me as i digged deeper into optics, but now I'm looking at E(k) vs k diagrams for semiconductors and this talk about k-space is confusing the heck out of me.

Maybe one should add, that in dispersion relations the k is proportional to the momentum carried by some particle or quasiparticle (crystal electron, phonon, exciton or others) and therefore E(k) diagrams are more or less plotting energy versus carried momentum.

 

[cmos]

Are you studying optics or solid state physics? The reason I ask is because there are some subtle differences in talking about the wavevector of an EM wave and the wavevector of an electron wave.

For waves in general, Redbelly gave the best physical explanation (post #2). When we start to talk E-k plots, these come as the result of performing Fourier analysis on the crystal lattice. The crystal lattice obviously exists in a position space. The Fourier transform of position is momentum which is directly proportional to the wavevector. In quantum mechanics, it is convenient to use wavevector rather than momentum. But really, when you talk about one, you equivalently talk about the other.

 

[James R]

The wavevector $\vec{k}$ has magnitude equal to the wave number $k = 2\pi / \lambda$ and a direction that is the same as the direction of propagation of the wave.

You can find the components of the wavevector using trigonometry. For example, in two dimensions the wavevector is:

$$\vec{k} = k_x \hat{i} + k_y \hat{j}$$

where

$$|\vec{k}| = \sqrt{k_x^2 +k_y^2}$$

and

$$k_x = |\vec{k}|cos \theta$$

where $\theta$ is the angle between $\vec{k}$ and the x axis, for example.

In three dimensions, things are a little more complicated, but you can still use trignometry.

Oh, and:

$${\nabla}^{2} U + {k}^{2} U = 0$$

expands as

$$\frac{\partial^2 U}{\partial x^2} + \frac{\partial^2 U}{\partial y^2} + \frac{\partial^2 U}{\partial z^2} + (k_x^2 + k_y^2 + k_z^2) U = 0$$,

which is, of course, a scalar equation, not a vector equation.

 

[harsimran singh]

hi ,
1ist of all try to understand physical meaning of physics that is what physics is
wave number :number of wavelengths(λ) present in 2pie radians
wavevector :it gives us direction of the wave in which it travels that,s why it is breakdown into components because u always cannot expect the wave to travel into one dimension .

please read intoduction to solid state physics by CHARLES KITTEL

 

[harsimran singh]

how i can ask questions ?

 

[Ertosthnes]

Start with something that makes sense, like the fact that you have more momentum with a smaller wavelength. So basically, $$p=h/\lambda,$$ where h is some constant.

Since $$\hbar=h/2\pi,$$
$$p=h/\lambda=2\pi\hbar/\lambda=\hbar k,$$
where $$k=2\pi/\lambda,$$ ie, k is just a vector in the direction of the momentum.