Energy Bands: Kronig-Penney Model

1. Feb 14, 2010

torehan

In Kronig-Penney Model we seek solutions for time independent Scrödinger equation inside and outside of the periodic potential.

for 0 < x < a;​
V(x) =
V0 for -b < x < 0​

My question is why solution of the KP Scrödinger equation in the region of $$-b < x < 0$$ doesnot have complex exponentials as in the region of $$0 < x < a$$ ?

2. Feb 14, 2010

Niles

In this region we have
$$\frac{{d^2 \psi }}{{dx^2 }} = k^2 \psi,$$

where

$$k^2 = \frac{{2m\left( {V_0 - E} \right)}}{{\hbar ^2 }}.$$
The solution to this 2nd order ODE is a linear combination of exponentials (with real arguments). If you think about it physically, the wavefunction will also decay in this area, which can only happen if the argument of the exponential is real.

3. Feb 14, 2010

torehan

OK, so what about first region?

$$\frac{{d^2 \psi }}{{dx^2 }} = q^2 \psi,$$

where
$$q^2 = \frac{{2m\left( {E} \right)}}{{\hbar ^2 }}.$$

Isn't it also 2.nd order ODE ?

4. Feb 14, 2010

Niles

In the region with V=0 we have
$$\frac{{d^2 \psi }}{{dx^2 }} = -\hat k^2 \psi,$$

where

$$\hat k^2 = \frac{{2m\left( {E} \right)}}{{\hbar ^2 }}.$$

Solutions are travelling waves (going left and right).

5. Feb 14, 2010

torehan

Oh i see, I had some conceptual errors on writing the wave eq.
Thanks!
Torehan

6. Feb 19, 2010

torehan

I know that this topic doesn't take much attention of most of you as there are more interesting topics about paradoxes of physics but I need a little bit guidance about reproducing the band structure diagrams shown in Kittel , 170p in 8th Edition :)

So we have different wave solution for V0 region and 0 region,

$$\psi_{1}(x)=Ae^{ikx}+Be^{-ikx}\ \ \ \ \ \ \ for \ \ \ \ \ \ \ \ 0<x<a$$

$$\psi_{2}(x)=Ce^{qx}+De^{-qx}\ \ \ \ \ \ \ for \ \ \ \ \ \ \ \ \-b<x<0$$

After we apply the boundary conditions (usual QM boundary condiions in in square well potentials ),

$$\psi_{1}(0) = \psi_{2}(0)$$ and $$\psi_{1}'(0) = \psi_{2}'(0)$$

$$\psi_{1}(a) = \psi_{2}(-b)$$ and $$\psi_{1}'(a) = \psi_{2}'(-b)$$

Then, we have four equations

$$A+B=C+D$$

$$ik(A-B)=q(C-D)$$

$$Ae^{ika}+Be^{-ika}=(Ce^{-qb}+De^{qb})e^{i\alpha(a+b)}$$

$$ik(Ae^{ika}-Be^{-ika})=q(Ce^{-qb}-De^{qb})e^{i\alpha(a+b)}$$

Determinant of the coefficients of this equation system mush vanish to have solutions. Than determinant yields,

$$([q^{2}-k^{2}]/2qk) sinh(qb)sin(ka) + cosh(qb)cosh(ka) = cos (\alpha(a+b))$$

Finally my issues
• How can I define the allowed and forbidden energy values?
• to be continued...

$$e^{i\alpha(a+b)}$$ ---->The term $$\alpha$$ came from Bloch's Theorem and (a+b) came from application of transitional symmetry operation to Bloch form wavefunction!

Last edited: Feb 19, 2010