What does the wavevector "k" mean in the Schrödinger eq. ?

[*Himanshu*]

why the solution for energy levels of electron in 1D crystal lattice as solved in Kronig penny model has used wave vector k differently then the Schrödinger equation solved for a free particle.
(only the conditions in the equation has changed not the maths...so the "USE" of wavevector 'k' must remain same and thus the physical significance)

Moreover I want to ask reason for below mentioned statements(source given at end)-
- >'We notice that exactly as in the case of the constant potential , the wave vector k has a twofold role: It is still a wave vector in the plane wave part of the solution, but also an index to yk(r) and uk(r) because it contains all the quantum numbers, which ennumerate the individual solutions.'

- >But in any case, the quantity k, while still being the wave vector of the plane wave that is part of the wave function (and which may be seen as the "backbone" of the Bloch functions), has lost its simple meaning: It can no longer be taken as a direct representation of the momentum p of the wave via p = [PLAIN]http://www.tf.uni-kiel.de/matwis/amat/semi_en/kap_2/illustr/h_quer.gif[I]k[/I][/B], or of its wavelength l = 2p/k, since:

The momentum of the electron moving in a periodic potential is no longer constant (as we will see shortly); for the standing waves resulting from (multiple) reflections at the Brillouin zones it is actually zero (because the velocity is zero), while k is not.

There is no unique wavelength to a plane wave modulated with some arbitrary (if periodic) function. Its Fourier decomposition can have any spectra of wavelengths, so which one is the one to associate with k?

To make this clear, sometimes the vector k for Bloch waves is called the "quasi wave vector".

Instead of associating k with the momentum of the electron, we may identify the quantity [PLAIN]http://www.tf.uni-kiel.de/matwis/amat/semi_en/kap_2/illustr/h_quer.gif[B][I]k[/I][/B], which is obviously still a constant, with the so-called crystal momentum P, something like the combined momentum of crystal and electron.

Whatever its name, [PLAIN]http://www.tf.uni-kiel.de/matwis/amat/semi_en/kap_2/illustr/h_quer.gif[B][I]k[/I][/B] is a constant of motion related to the particular wave yk(r) with the index k. Only if V = 0, i.e. there is no periodic potential, is the electron momentum equal to the crystal momentum; i.e. the part of the crystal is zero.

The crystal momentum P, while not a "true" momentum which should be expressible as the product of a distinct mass and a velocity, still has many properties of momentums, in particular it is conserved during all kinds of processes.----As mentioned at -- http://www.tf.uni-kiel.de/matwis/amat/semi_en/kap_2/backbone/r2_1_4.html

 

[Simon Bridge]

Welcome to PF;

why the solution for energy levels of electron in 1D crystal lattice as solved in Kronig penny model has used wave vector k differently then the Schrödinger equation solved for a free particle.

... well those are different situations.
But how has the wave vector been used differently?

You can use any characteristic of a mode as a way to label (index) the mode.

 

[tijana]

For free particles. Wave function is flat wave given by wave number k: where k²=2mE/ћ²

 

[Simon Bridge]

Good point @tijana ... specifically, for free particles, the wave-vector can take on much any value, which makes it useless as a general index (although we can use the wave-vectors of two different waves as labels to distinguish them). In some constraining systems, though, the wave-vector must take on discrete values ... so it can be used as an index for the state rather than just the name of a wave.

Some character of a function being used as a label in no way changes how it is used in physics, the labels just help us have discussions.