science forum beginner
Joined: 26 Jun 2006
|Posted: Mon Jun 26, 2006 10:07 pm Post subject:
Linear momentum operator in spherical coordinates in non relativistic QM
In non relativistic quantum mechanics the radial component of momentum
operator in spherical coordinates is (see, e.g. Messiah ch. IX):
p = ( d/dr + 1/r)
(here and in the following p means p_r, d means partial derivative, h
means h/(2pi) and f(r) is a wave function in spherical coordinates with
the argument r a vector)
How does this expression fit with the fact that momentum is the
generator of translations?
In fact, in spherical coordinates, for an infinitesimal displacement a
along the direction of r, we have for a wave function f(r) :
f(r-a) = f(r) -a df/dr + O(a^2)
and that's not equal to
exp(-i/h ap) f(r) = f(r) -a df/dr -a f/r + O(a^2)
Thanks in advance for your advice.
I'll be grateful also if you point me to some (not much advanced!)
clarifying literature on this subject.