Michael Fowler 10/29/07 As a warm up to analyzing how a wave
function transforms under rotation, we review the effect of
linear translation on a single particle wave function . We have
already seen an example of this: the coherent states of a simple
harmonic oscillator discussed earlier were (at t = 0) identical
to the ground state except that they were centered at some point
displaced from the origin. In fact, the operator creating such a
state from the ground state is a translation operator. The
translation operator T(a) is defined at that operator which when
it acts on a wave function ket gives the ket corresponding to
that wave function moved over by a, that is, so, for example, if
is a wave function centered at the origin, T(a) moves it to be
centered at the point a. We have written the wave function as a
ket here to emphasize the parallels between this operation and
some later ones, but it is simpler at this point to just work
with the wave function as a function, so we will drop the ket
bracket for now