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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?

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