Dispersion relation (particle in a box)

[HAMJOOP]

I am learning some basic solid state physics idea, like density of state ...etc.

For particle in a 1D box,

E = n^2 (pi)^2 (h_bar)^2 / 2mL^2

But why it is written as
E = (h_bar)^2 k^2 /2m

does it means that energy eigenvalue E is related to momentum k ?
I guess it is not because momentum is not eigenstate.

But what is this expression talking about anyway ? what is the physical meaning of this k ?

 

[DrDu]

While p isn't an eigenstate, p^2 is. So take k^2 as eigenvalue of p^2. In solid state physics the point is that the energy spectrum in the large volume limit becomes independent of the boundary conditions. So you could also assume periodic boundary conditions and interpret k as momentum.