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Forum index » Science and Technology » Physics » Particle
Linear momentum operator in spherical coordinates in non relativistic QM
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Laura B
science forum beginner


Joined: 26 Jun 2006
Posts: 1

PostPosted: Mon Jun 26, 2006 10:07 pm    Post subject: Linear momentum operator in spherical coordinates in non relativistic QM Reply with quote

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.


Laura B.
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