Exploring Angular Momentum and the HUT: Eigenstates and Uncertainty

[broegger]

Hi,

The operators representing the components of angular momentum are incompatible, since

$$[L_x,L_y] = i\hslash L_z$$.
​

If you apply the uncertainty principle to this you get:

$$\sigma_{L_x}\sigma_{L_y} \geq \tfrac{\hslash}2|\langle L_z \rangle|$$.
​

But what if $$\langle L_z \rangle = 0$$? Then the HUT does not prevent simultanoeus eigenstates of Lx and Ly or what?

 

[dextercioby]

Yes,it's true.U can,under certain conditions,measure eigenvalues of $\hat{L}_{x}$ and $\hat{L}_{y}$ exactly...

I think you meant

$$\left[\hat{L}_{x},\hat{L}_{y}\right]_{-}=i\hbar\hat{L}_{z}$$

Daniel.

 

[broegger]

Yes,it's true.U can,under certain conditions,measure eigenvalues of $\hat{L}_{x}$ and $\hat{L}_{y}$ exactly...

Oh, could you give a quick example? My book doesn't mention this: it just states that it is impossible to have simultaneous eigenstates of Lx and Ly because they don't commute. Everytime <Lz>=0 (like the ground state of hydrogen, right?) the HUT allows simultaneous eigenstates of Lx and Ly, but is it very uncommon (since my book doesn't mention it)?

I think you meant

$$\left[\hat{L}_{x},\hat{L}_{y}\right]_{-}=i\hbar\hat{L}_{z}$$

Daniel.

Yes, edited.

 

[dextercioby]

Oh, could you give a quick example?

I think you found it yourself.H atom in the fundamental state (1s) has

$$\langle\hat{L}_{z}\rangle_{|1,0,0\rangle}=0$$

My book doesn't mention this: it just states that it is impossible to have simultaneous eigenstates of Lx and Ly because they don't commute.

This means that they don't analyze the cases of the general uncertainty relations...That's bad... :yuck:

Everytime <Lz>=0 (like the ground state of hydrogen, right?) the HUT allows simultaneous eigenstates of Lx and Ly, but is it very uncommon (since my book doesn't mention it)?

Why is it very uncommon.I'd say the H atom is very common...~75% of solar mass is hydrogen...


Daniel.

 

[broegger]

Yeah, the H-atom ground state has <Lz>=0 but that doesn't necessarily imply that it is a simultaneous eigenstate of Lx and Ly: the HUT is an inequality, $$\sigma_{L_x}\sigma_{L_y} \geq 0$$, so I don't know.

 

[dextercioby]

Compute

$$\hat{L}_{x}|1,0,0\rangle$$

$$\hat{L}_{y}|1,0,0\rangle$$

$$\hat{L}_{z}|1,0,0\rangle$$

And then the uncertainties...And verify the uncertainty relation...

Daniel.