(adsbygoogle = window.adsbygoogle || []).push({}); Using the Schrodinger equation for an electron in a periodic potential where U(r +R) [R is the translation vector R=n1a1+n2a2+n3a3 and ni are intergers and ai are teh primitive lattice vectors, G is for reciprocal lattice G=n1b1+n2b2+n3b3 and ni are intergers and bi= (2PI*aj x ak)/(a1 . a2 x a3)]

a)Show that the periodic potential can be expanded as

U(r)=SUM (over G) exp(iG.r) . U(G)

show the potential is real and is reflection symmetric U(-r)=U(r)

show that implies U(-G)=U(complex conjiguate)(G)=U(G)

potential is chosen as U(G=0)=0

2. Relevant equations

I hope you know the SE for a periodic potential......

3. The attempt at a solution

can i jump straight to the fourier transform for the U(r+R)

f(x)=SUM(over m) exp(imx)f(m)

bcause after that it's just loosing the

exp (iR.G)=1

I don't get the complex conjugation of U(r) = U(r) unless the "i" in the exponential isn't changed then there wouldn't be a complex part......

U(-r) = U(r) cos it's jsut a translation in the real lattice which is stated in teh question.

It implies that U(-G)=U(complex conjiguate)(G)=U(G) cos the form of U(r) has U(G) in it

**Physics Forums | Science Articles, Homework Help, Discussion**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Homework Help: Bandgap in a weak periodic potential

Can you offer guidance or do you also need help?

Draft saved
Draft deleted

**Physics Forums | Science Articles, Homework Help, Discussion**