The adiabatic approximation H dependents on a set of parameters Changed adiabatically H(R()平=E(RO)F) En(R(0)→>En(R(t) Hn(R(O)→|平n(R() The adiabatic theorem If the system is initially in a non-degenerate energy eigenstate, it will remain in the corresponding eigenstate during the whole adiabatic evolution 16
16 The adiabatic approximation • H dependents on a set of parameters • Changed adiabatically The adiabatic theorem: If the system is initially in a non-degenerate energy eigenstate, it will remain in the corresponding eigenstate during the whole adiabatic evolution. H(R(t)) = E(R(t)) ( (0)) ( ( )) ( (0)) ( ( )) R R t E R E R t n n n n → →
The Berry phase M.V. Berry(1984) a closed path in parameter space: R(T)=R(O) The initial state is one of energy eigenstates The final state differs from the initial one by a phase factor 1(n(C)+yn) (0 Where Dynamic phase ,E,(R()t Berry phase y,(C)=14、R)VRn(R)dR 17
17 The Berry phase • A closed path in parameter space: R(T) = R(0) • The initial state is one of energy eigenstates • The final state differs from the initial one by a phase factor Where • Dynamic phase • Berry phase ( ) (0) ( ( ) ) d n C n i T e + = = − T n d n E R t dt i 0 ( ( )) C i n R n R dR C n R = ( ) ( ) ( ) M. V. Berry (1984)