Timeline for Orbital angular momentum as sum of harmonic oscillators
Current License: CC BY-SA 4.0
15 events
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| Apr 13, 2021 at 0:02 | vote | accept | dnvlz | ||
| Apr 12, 2021 at 18:03 | comment | added | Qmechanic♦ | Ballentine's method is proven in my Phys.SE answer here. | |
| Apr 12, 2021 at 17:15 | history | edited | Qmechanic♦ | CC BY-SA 4.0 |
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| Apr 12, 2021 at 16:58 | comment | added | ytlu | The algebraic structure of $L_+$ and $L_-$ is similar to that of $a^\dagger$ and $a$ in the harmonic oscillator. This is a smart device to put this two systems into a similar form of Hamiltonian. But it doesn't imply they have same eigen function in r-space. | |
| Apr 12, 2021 at 16:39 | comment | added | ytlu | The Hermite polynomial is starting from the ground state of oscillator: $a | 0 \rangle=0$, translating into r-space differential equation $\ x \psi_0 + \frac{\partial \psi_0}{\partial x} = 0$ with proper scaling. The r-space representation of the $\frac{1}{2}(p_1^2 + q_1^2)$ is different from the harmonic oscillator, therefore, it eigen function in r-space will not be Hermite polynomial. | |
| Apr 12, 2021 at 15:39 | answer | added | Cosmas Zachos | timeline score: 5 | |
| Apr 12, 2021 at 14:14 | comment | added | dnvlz | @user1379857 Do you mean $n_1-n_2$ can be any integer since $n_i \in \mathbb{N}$? Would that imply there is no highest (or lowest) weight state as in the case $Y_l^m$ where $m\leq l$ (and $m\geq -l$)? | |
| Apr 12, 2021 at 3:12 | history | edited | dnvlz | CC BY-SA 4.0 |
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| Apr 12, 2021 at 3:01 | history | edited | dnvlz | CC BY-SA 4.0 |
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| Apr 12, 2021 at 2:51 | comment | added | Cosmas Zachos | Linked. | |
| Apr 12, 2021 at 2:20 | comment | added | user1379857 | Hmm... In order to construct representations of $SO(3)$, you define the raising/lowering operators $L_\pm = L_1 \pm i L_2.$ You then solve for a highest weight state that satisfies $L_3 | l, l \rangle = l |l, l \rangle$ and $L_+ | l, l \rangle = 0$. You then lower that state $l$ times using $L_-$ to get the rest of the vectors in the representation. What happens when you try to do that here? The special property of spherical harmonics is that they satisfy $L^2 Y_{lm} = l(l+1) Y_{lm}. $ Is that the case here? | |
| Apr 12, 2021 at 2:17 | review | Close votes | |||
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| Apr 12, 2021 at 0:32 | history | edited | dnvlz | CC BY-SA 4.0 |
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| Apr 12, 2021 at 0:26 | review | First posts | |||
| Apr 12, 2021 at 1:40 | |||||
| Apr 12, 2021 at 0:23 | history | asked | dnvlz | CC BY-SA 4.0 |