- #1
tim_lou
- 682
- 1
for the solution to the time-independent Schrödinger's equation with a period potential,
[tex]V(x)=V(x+a)[/tex]
one has:
[tex]\psi(x)=Af(x)+Bg(x)[/tex]
and
[tex]\psi(x+a)=A'f(x)+B'g(x)[/tex]
the coefficients are related by a matrix equation,
let
[tex]v=[A, B]^T[/tex]
[tex]v'=[A', B']^T[/tex]
then
[tex]v=Kv'[/tex]
where K is some non-singular matrix.
hence we can choose a solution where v is the eigenvector to the matrix K, more specifically,
[tex]v'=Kv=\lambda v[/tex]
and thus one can find solutions to the Schrödinger's equation that satisfy:
[tex]f(x+a)=\lambda f(x)[/tex] (1)
from there, one can show that there are band structures for the allowed energies.
However, how does one prove that these are the ONLY physical solution to the Schroedinger's equation? I mean without the constrain of (1), we can possibly have other wave functions with different energies (in the forbidden zone)? am I missing something?
[tex]V(x)=V(x+a)[/tex]
one has:
[tex]\psi(x)=Af(x)+Bg(x)[/tex]
and
[tex]\psi(x+a)=A'f(x)+B'g(x)[/tex]
the coefficients are related by a matrix equation,
let
[tex]v=[A, B]^T[/tex]
[tex]v'=[A', B']^T[/tex]
then
[tex]v=Kv'[/tex]
where K is some non-singular matrix.
hence we can choose a solution where v is the eigenvector to the matrix K, more specifically,
[tex]v'=Kv=\lambda v[/tex]
and thus one can find solutions to the Schrödinger's equation that satisfy:
[tex]f(x+a)=\lambda f(x)[/tex] (1)
from there, one can show that there are band structures for the allowed energies.
However, how does one prove that these are the ONLY physical solution to the Schroedinger's equation? I mean without the constrain of (1), we can possibly have other wave functions with different energies (in the forbidden zone)? am I missing something?
Last edited: