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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

[tex]\overrightarrow{k} = (k_x , k_y , k_z)[/tex]

where wavenumber k is

[tex] k = | \overrightarrow{k} | = \sqrt{{k_x}^2 + {k_y}^2 + {k_z}^2}[/tex]

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 [tex] k= \frac{2 \pi}{ \lambda} [/tex]

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

[tex] {\nabla}^{2} U + {k}^{2} U = 0 [/tex] 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

[tex] \frac{{\partial}^{2}}{{\partial x}^{2}} \left ( U (x,t) \right ) + {k_x}^{2} U(x,t) =0 [/tex]

[tex] \frac{{\partial}^{2}}{{\partial y}^{2}} \left ( U (y,t) \right ) + {k_y}^{2} U(y,t) =0 [/tex]

[tex] \frac{{\partial}^{2}}{{\partial z}^{2}} \left ( U (z,t) \right ) + {k_z}^{2} U(z,t) =0 [/tex]

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 alsoi simply just don't understand why they break up the wavevector into componentswhose magnitude is the wave number which is clearly (and understandably) defined as inversely proportional to the wavelength

thanks guys.

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# What is wavevector?

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