▁mechanical ▁Hamilton ians ▁are ▁time ▁dependent . ▁How ▁to ▁solve ▁problems ▁where ▁there ▁is ▁an ▁explicit ▁time ▁dependence ▁is ▁an ▁open ▁subject ▁now ad ays . ▁For ▁problems ▁of ▁this ▁kind ▁it ▁is ▁of ▁importance ▁to ▁look ▁for ▁constants ▁of ▁motion ▁or ▁invari ants . ▁For ▁the ▁( time ▁dependent ) ▁harm onic ▁oscill ator ▁it ▁is ▁possible ▁to ▁write ▁several ▁invari ants , ▁among ▁them , ▁the ▁Er m akov – L ew is ▁invariant ▁which ▁is ▁developed ▁below . ▁ ▁The ▁time ▁dependent ▁harm onic ▁oscill ator ▁Hamiltonian ▁reads ▁▁▁ ▁It ▁is ▁well ▁known ▁that ▁an ▁invariant ▁for ▁this ▁type ▁of ▁interaction ▁has ▁the ▁form ▁▁▁ ▁where ▁ ▁ob e ys ▁the ▁Er m akov ▁equation ▁ ▁The ▁above ▁invariant ▁is ▁the ▁so - called ▁Er m akov – L ew is ▁invariant . ▁It ▁is ▁easy ▁to ▁show ▁that ▁ ▁may ▁be ▁related ▁to ▁the ▁time ▁independent ▁harm onic ▁oscill ator ▁Hamiltonian ▁via ▁a ▁unit ary ▁transformation ▁of ▁the ▁form ▁▁ ▁as ▁ ▁This ▁allows ▁an ▁easy ▁form ▁to ▁express ▁the ▁solution ▁of ▁the ▁Sch r öd inger ▁equation ▁for ▁the ▁time ▁dependent ▁Hamiltonian . ▁ ▁The ▁first ▁exponential ▁in ▁the ▁transformation ▁is ▁the ▁so - called ▁s que e ze ▁operator . ▁ ▁This ▁approach ▁may ▁allow ▁to ▁simplify ▁problems ▁such ▁as ▁the ▁Qu ad ru pole ▁ion ▁trap , ▁where ▁an ▁ion ▁is ▁tra pped ▁in ▁a ▁harm onic ▁potential ▁with ▁time ▁dependent ▁frequency . ▁The ▁transformation ▁presented ▁here ▁is ▁then ▁useful ▁to ▁take ▁into ▁account ▁such ▁effects