# Wave vector

In physics , the wave vector or wave number vector is a vector that is perpendicular to the wave front of a wave and is its magnitude  , where is the wavelength . The unit of measurement for the components is 1 / m. In most cases it indicates the direction of propagation of the wave, but the direction of the Poynting vector for the flow of energy in electromagnetic waves in certain media can deviate from the wave vector. ${\ displaystyle {\ vec {k}}}$${\ displaystyle {\ frac {2 \ pi} {\ lambda}}}$${\ displaystyle \ lambda}$

## description

A plane wave propagating in the direction can be written as: ${\ displaystyle {\ vec {k}}}$

${\ displaystyle \ psi ({\ vec {r}}, t) = A \ cdot e ^ {i ({\ vec {k}} \ cdot {\ vec {r}} - \ omega t)}}$

With

• amplitude ${\ displaystyle A}$
• Euler's number ${\ displaystyle e}$
• imaginary unit ${\ displaystyle i}$
• Position vector ${\ displaystyle {\ vec {r}}}$
• Angular frequency ${\ displaystyle \ omega}$
• Time .${\ displaystyle t}$

With the components in the x, y and z directions

${\ displaystyle {\ vec {k}} = (k_ {x}, k_ {y}, k_ {z})}$

shows the wave vector in 3-dimensional k-space, also called reciprocal space , in a certain direction.

The magnitude of the wave vector is the circular wave number , hence the name wave number vector: ${\ displaystyle k}$

${\ displaystyle k = | {\ vec {k}} | = {\ frac {\ omega} {c}} = {\ frac {2 \ pi} {\ lambda}},}$

in which

• ${\ displaystyle c}$the phase velocity and
• ${\ displaystyle \ lambda}$the wavelength is.

## Wave vector and quantum numbers

Without further boundary conditions , for example in a vacuum, the wave vector of a particle can continuously assume any amount and any orientation. However, under certain circumstances the wave vector is a quantized quantity.

The restriction of particles to a finite space, for example in a potential well or the lattice of a solid , means that the steady state of the system can only assume discrete values. In this case, the wave vector is quantized, even if, strictly speaking, it does not represent quantum numbers. Rather, the wave vector is a function of quantum numbers, or its possible values ​​can be counted using quantum numbers. This can be seen in analogy to the intrinsic energies of a quantum mechanical problem with a discrete spectrum : the index of the discrete energy is the quantum number, but not the energy itself. ${\ displaystyle E_ {n}}$${\ displaystyle n}$

Example: For the solutions of the Schrödinger equation of a three-dimensional, infinitely high potential well, the edge lengths apply ${\ displaystyle a}$

${\ displaystyle \ Psi _ {n_ {x}, n_ {y}, n_ {z}} (x, y, z) \; = \; A \, \ sin \ left (k_ {x} \, x \ right) \, \ sin \ left (k_ {y} \, y \ right) \, \ sin \ left (k_ {z} \, z \ right)}$

with the amplitude and the abbreviation ${\ displaystyle A}$

${\ displaystyle k_ {i} = {\ frac {n_ {i} \, \ pi} {a}}}$.

It is a non-negative integer and the index may have the values , or accept. ${\ displaystyle n_ {i}}$${\ displaystyle i}$${\ displaystyle x}$${\ displaystyle y}$${\ displaystyle z}$

The stationary states of the particle, so by the quantum numbers , and characterized. Instead of naming a state using this number triple, the wave vector can also be used. However, the wave vector or one of its components must not be referred to as a quantum number, because on the one hand it is dimensional and on the other it is represented by real numbers . ${\ displaystyle n_ {x}}$${\ displaystyle n_ {y}}$${\ displaystyle n_ {z}}$${\ displaystyle {\ vec {k}} = (k_ {x}, k_ {y}, k_ {z})}$

A potential well with particles results in vectors in reciprocal space . When it comes to fermions , there are only a limited number of stationary states per wave vector. The number of these results from the amount of spin of the considered particles. Electrons are particles for which the amount of spin has the value . Such a spin can only assume two orientations with respect to a quantization axis . Therefore each wave vector in the potential well can be assumed to have a maximum of two electrons. ${\ displaystyle N}$${\ displaystyle N}$${\ displaystyle 1/2}$

## Wave vector and momentum

With photons ( Einstein equations ) as well as with matter waves (De Broglie relation) the vectorial momentum is proportional to the wave vector, with the reduced Planck constant as the proportionality factor: ${\ displaystyle {\ vec {p}}}$ ${\ displaystyle \ hbar}$

${\ displaystyle {\ vec {p}} = \ hbar {\ vec {k}}}$

## literature

• Charles Kittel: Introduction to Solid State Physics . 15th edition, Oldenbourg Verlag Munich, Munich 2013, ISBN 978-3-486-59755-4 .
• Bahaa EA Saleh, Malvin Carl Teich: Fundamentals of Photonics. 1st edition, Wiley-VCH Verlag GmbH & Co, Weinheim 2008, ISBN 978-3-527-40677-7 .