Our states will have positive energy.
This potential has a new symmetry, that atranslation by the lattice spacingaleaves the
problem unchanged. The probability distributions must therefore have this symmetry
|ψ(x+a)|^2 =|ψ(x)|^2 ,which means that the wave function differs by a phase at most.
ψ(x+a) =eiφψ(x)The general solution in the region (n−1)a < x < nais
ψn(x) =Ansin(k[x−na]) +Bncos(k[x−na])k=√
2 mE
̄h^2By matching the boundary conditions and requiring that the probability be periodic, we derive a
constraint onk(See section 9.7.6) similar to the quantized energies for bound states.
cos(φ) = cos(ka) +maV 0
̄h^2 ksin(ka)Since cos(φ) can only take on values between -1 and 1, there areallowed bands ofkand gaps
between those bands.
The graph below shows cos(ka) +maV ̄h (^2) k^0 sin(ka) as a function ofk. If this is not between -1 and 1,
there is no solution, that value ofkand the corresponding energy are not allowed.
Energy Bands
-2
-1
0
1
2
0510
k (/A)
cos(phi)
cos(phi)
E