Periodic potential & Harmonic oscillator 2011.09.27 何峰
Periodic potential & Harmonic oscillator 2011.09.27 何 峰
Periodic potential V(x+na=V(x)n=0,±1，±2，…
Periodic potential V(x na) V(x),n 0,1,2,
Periodicity 会品-mn-E 式可"ea-o小-5u-w +r闲lr+n-st+a .'.(x+a),(x)are all solutions of the same hamiltonian
Periodicity ( ), ( )are allsolutions of thesame hamiltonian. ( ) ( ) ( ) 2 ( ) ( ) ( ) 2 2 2 2 2 2 2 x a x V x x a E x a dx d m V x a x a E x a d x a d m ( ) ( ) ( ) 2 2 2 2 V x x E x dx d m
If there is no degeneracy w(x+a)=Cw(x) w(x+na)=C"w(x) y(x)=C"v(x+na) C≡l→C=ea
If there is no degeneracy (x a) C(x) (x na) C (x) n (x) C (x na) n n i C 1C e
If there is degeneracy U1(x+a)=C1U1(x)+C12U2(x) U2(x+a)=C2U1(x)+C22U2(x) Do we have P(x+a)=(x)? Construct the wavefunction Ψ(x)=AU1(x)+BU2(x) |8c 二0 C12 Ψ1(x+a)=Ψ1(x) Ψ2(x+a)=2Ψ2(x)
If there is degeneracy ( ) ( ) ( ) ( ) ( ) ( ) 2 2 1 1 2 2 2 1 1 1 1 1 2 2 U x a C U x C U x U x a C U x C U x Do we have (x a) (x) ? ( ) ( ) ( ) ( ) 2 2 2 1 1 1 x a x x a x 0 1 2 2 2 1 1 2 1 C C C C ( ) ( ) ( ) Construct the wavefunction 1 2 x AU x BU x