I'm trying to show the relation between L^2 and Lz where L is total angular momentum and Lz is the z component.(adsbygoogle = window.adsbygoogle || []).push({});

Given f is an eigenfunction of both L^2 and Lz

L^2f = [lamb] f Lz f = [mu] f and L^2 = Lx^2 + Ly^2 + Lz^2 then

<L^2> = < Lx^2 + Ly^2 + Lz^2> = <Lx^2> + <Ly^2> + <Lz^2>

<L^2> = [inte]f* L^2 f dr = [inte]f* [lamb] f dr

= [lamb] [inte]f*fdr = [lamb] <f> = [lamb]

<Lz> = [inte]f* Lz^2 f dr = [inte]f* [mu]^2 f dr = [mu]^2 [inte]f*f dr

= [mu]^2

Now

<lz^2> = <L^2> - <Lx^2> - <Ly^2> substituting gives

[mu]^2 = [lamb] - <Lx^2> - <Ly^2>

or [mu] squared is less than or equal to [lamb] with equality only when <Lx^2> = <Ly^2> = zero

I would like to know if the above is correct and is this just another way of saying that Lz^2 <= to L^2 ?

**Physics Forums - The Fusion of Science and Community**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Eigenvalues of L^2 and Lz^2

Loading...

Similar Threads - Eigenvalues Lz^2 | Date |
---|---|

I Two dimenstional Heisenberg Hamiltonian for spin 1/2 system | Tuesday at 2:16 PM |

I Probabilities for degenerate eigenvalues? | Jan 29, 2018 |

I Eigenvectors - eigenvalues mappings in QM | Jan 1, 2018 |

I Pauli Spin Operator Eigenvalues For Two Electron System | Aug 26, 2017 |

I How do i find the eigenvalues of this tough Hamiltonian? | Apr 8, 2017 |

**Physics Forums - The Fusion of Science and Community**