Conservation of angular momentum (central force)

[HAMJOOP]

In a central force problem,
angular momentum is conserved.
we quantized one of the component of L, say Lz.
Also, we quantized the angular momentum, L = √l(l+1)h_bar
If we know Lx and Ly without uncertainty,
then we know the direction of L.

Hence we know the motion of the particle is confined in a plane(let say x-y plane).
Then we know exactly the position in z direction, which contradicts the uncertainty principle.Hence, we can't know Lx and Ly without uncertainty.

Does that mean the direction of the angular momentum is uncertain ?
So, can we say angular momentum is conserved ?One more Thing (particle in a box)
Can I say E = (p^2 /2m) ?
coz E is quantized, so quantization of p violates the uncertainty principle.
What's wrong?

 

[The_Duck]

Hence, we can't know Lx and Ly without uncertainty.

You can already get this from the fact that Lx and Ly don't commute, so it's impossible to have a state with definite values of both Lx and Ly.

Does that mean the direction of the angular momentum is uncertain ?

Yes.

So, can we say angular momentum is conserved ?

Yes. Lx, Ly, and Lz are all conserved quantities. In QM this essentially means that the probability distribution of the possible values of Lx (say) does not change with time.

 

[HAMJOOP]

In classical mechanics, we say angular momentum is conserved when its magnitude and direction are constant for all time.

But in quantum mechanics, the direction of angular momentum is uncertain.
So, can I say its direction is changing all the time ??

 

[Nugatory]

But in quantum mechanics, the direction of angular momentum is uncertain.
So, can I say its direction is changing all the time ??

You can say it, but that doesn't make it right

It would be better to say that it doesn't have a direction until you've measured it (that's not really right either, but it's much less likely to lead you astray when you come to some of the more difficult problems of interpreting QM. You might want to take a look at this web page for more background).