Translational factor 1 Single atom Consider an atom moving in space with velocity v=dR/dt. The object is to solve the time dependent Schroedinger equation i H t (1) where H is the electronic Hamiltonian 2r H V . 2 (2) Assume that we know the time independent solutions n(r) at R=0 such that Hn(r)=Enn(r). (3) Try a solution of the form (4) r , R ,t exp i E n n t f r , R ,t n r R The relationship (5) 2 ln ln ln 2 ln so that 2 f f ln f ln f f 2 ln f (6) f ln f ln f 2 ln f ln ln ln 2 ln 2 ln f enables us to write eqn. 1 as ln f En n i iv R ln f iv R ln n r , R , t t (7) 1 2 ln f r ln f r ln f 2 r ln f r ln 2 V r , R , t 2 ln r ln r ln from which we can collect the terms involving on both sides of the equation to eliminate the potential and the En yielding. ln f n i iv R ln f iv R ln n r , R , t t 1 2 ln f r ln f r ln f 2 r ln f r ln r , R , t 2 Try the form f=exp(ik(t)r), and note that R=-r and Rf=0 k 1 r iv r ln n r , R ,t k k i 2 k r ln r , R ,t n t 2 which solves the equation provided k=v=dR/dt and n=k2/2. Thus for a single atom with a single electron moving with constant velocity v, the electronic wave function is 2 1 dR dR r , R ,t expi E n t exp r n r R t 2 dt dt