Angular momentum in whole numbers

[snoopies622]

The dimension of Planck's constant h (ML²/T) is also the dimension of angular momentum. Does it follow that the angular momentum of any object must be hn where n is an integer? I know h was discovered in a different context, but I was just wondering.

 

[Meir Achuz]

It follows from quantum mechanics, not just the units, that L=nh.
For a classical object, n is so large that L appears continuous.

 

[snoopies622]

Here's the problem I'm having - consider the following scenario:

Using a standard Cartesian coordinate system, a particle of mass m moves along the x-axis in the positive x direction with constant speed v.

Using (0,y) as a reference point (and assuming that v<<c) the scalar value of the particle's angular momentum is ymv. If angular momentum is always hn - where n is an integer - it follows that the particle's speed must be hn/ym.

Since our reference point is arbitrary we can change it to 2y and this would make the allowable speeds of the particle be hn/2ym, which of course includes speeds not included in the set hn/ym.

Since nothing has changed with the particle, why would it now have a different set of allowable speeds?

 

[alxm]

Angular momentum would be in terms of h-bar, rather.

 

[Meir Achuz]

Yes, it should be hbar, but I was using units with 2pi=1.
Snoopie, your problem is you are trying to use a classical impact parameter with a quantum trajectory. There is no fixed y.

 

[snoopies622]

...you are trying to use a classical impact parameter with a quantum trajectory. There is no fixed y.

So angular momentum, action, or any other physical quantity with that dimension is precisely defined while all others (distance, energy, etc.) are not?

 

[atyy]

Orbital angular momentum, spin angular momentum. Each are defined by the commutation relations. Spin can be half integer.

 

[snoopies622]

I'm sorry; I guess I don't really understand even the most basic ideas of quantum mechanics - I thought I did.

Is it the case that y should be replaced with y + Δy and (if the mass m is known with certainty) v should be replaced by v + Δv such that

$$m \Delta y \Delta v = \hbar$$

while the angular momentum L = m (y + Δy) (v + Δv) is still known to be exactly $n \hbar$ ?

 

[Meir Achuz]

No, its more detailed than that. The particle moving is like an ocean wave. How would you describe the position of a wave? You need to look at a simple QM text.

 

[snoopies622]

You need to look at a simple QM text.

Is there a specific one you have in mind?

 

[atyy]

$$m \Delta y \Delta v = \hbar$$

That's the heuristic uncertainty principle. The formal uncertainty principle is given by the commutation relations which look something like [p,x]=i# (I'm making it up, the correct ones should be in standard texts - maybe try Chapter 3 of Peebles's, or one of 't Hooft's recommendations: http://quantummechanics.ucsd.edu/ph130a/130_notes/node10.html, http://quantummechanics.ucsd.edu/ph130a/130_notes/node22.html).

 

[snoopies622]

Thanks atyy. You always provide great resources!