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
Hn(r)=Enn(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   expi E n    t  exp  r n r  R t 
2  dt     dt

 

