130_notes.dvi

This energy band phenomenon is found in solids. Solids with partially filledbands are conductors.
Solids with filled bands are insulators. Semiconductors have a small number of charge carriers (or
holes) in a band.

9.6 The Quantum Rotor

It is useful to simply investigateangular momentum with just one free rotation angle. This
might be the quantum plane propeller. We will do a good job of this in 3 dimensions later.

Lets assume we have amassmconstrained to move in a circle of radiusr. Assume the
motion in the circle is free, so there is no potential. The kinetic energyis^12 mv^2 = p

2

2 mforp=mr

dφ

dt.

If we measure distance around the circle, thenx=rφand the one problem we have is that once I go
completely around the circle, I am back tox= 0. Lets just go ahead and write our wavefunction.

ei(px−Et)/ ̄h=ei(prφ−Et)/ ̄h

Remembering angular momentum, lets call the combinationpr=L. Our wave isei(Lφ−Et)/ ̄h.

This must besingle valuedso we need to require that

ei(2πL−Et)/h ̄=ei(0−Et)/ ̄h

ei(2πL)/ ̄h= 1

L=n ̄h