Salvato in:
| Autori principali: | , , |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2023
|
| Soggetti: | |
| Accesso online: | https://arxiv.org/abs/2305.11456 |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866909128290467840 |
|---|---|
| author | Rakitzis, T. Peter Koutrakis, Michail E. Katsoprinakis, George E. |
| author_facet | Rakitzis, T. Peter Koutrakis, Michail E. Katsoprinakis, George E. |
| contents | In quantum mechanics, spatial wavefunctions describe distributions of a particle's position or momentum, but not of angular momentum $j$. In contrast, here we show that a spatial wavefunction, $j_m (ϕ,θ,χ)=~e^{i m ϕ} δ(θ- θ_m) ~e^{i(j+1/2)χ}$, which treats $j$ in the $|jm>$ state as a three-dimensional entity, is an asymptotic eigenfunction of angular-momentum operators; $ϕ$, $θ$, $χ$ are the Euler angles, and $cos θ_m=(m/|j|)$ is the Vector-Model polar angle. The $j_m (ϕ,θ,χ)$ gives a computationally simple description of particle and orbital-angular-momentum wavepackets (constructed from Gaussian distributions in $j$ and $m$) which predicts the effective wavepacket angular uncertainty relations for $Δm Δϕ$, $Δj Δχ$, and $ΔϕΔθ$, and the position of the particle-wavepacket angular motion on the orbital plane. The particle-wavepacket rotation can be experimentally probed through continuous and non-destructive $j$-rotation measurements. We also use the $j_m (ϕ,θ,χ)$ to determine well-known asymptotic expressions for Clebsch-Gordan coefficients, Wigner d-functions, the gyromagnetic ratio of elementary particles, $g=2$, and the m-state-correlation matrix elements, $<j_3 m_3|j_{1X} j_{2X}|j_3 m_3>$. Interestingly, for low j, even down to $j=1/2$, these expressions are either exact (the last two) or excellent approximations (the first two), showing that $j_m (ϕ,θ,χ)$ gives a useful spatial description of quantum-mechanical angular momentum, and provides a smooth connection with classical angular momentum. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2305_11456 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | The Vector-Model Wavefunction: spatial description and wavepacket formation of quantum-mechanical angular momenta Rakitzis, T. Peter Koutrakis, Michail E. Katsoprinakis, George E. Quantum Physics In quantum mechanics, spatial wavefunctions describe distributions of a particle's position or momentum, but not of angular momentum $j$. In contrast, here we show that a spatial wavefunction, $j_m (ϕ,θ,χ)=~e^{i m ϕ} δ(θ- θ_m) ~e^{i(j+1/2)χ}$, which treats $j$ in the $|jm>$ state as a three-dimensional entity, is an asymptotic eigenfunction of angular-momentum operators; $ϕ$, $θ$, $χ$ are the Euler angles, and $cos θ_m=(m/|j|)$ is the Vector-Model polar angle. The $j_m (ϕ,θ,χ)$ gives a computationally simple description of particle and orbital-angular-momentum wavepackets (constructed from Gaussian distributions in $j$ and $m$) which predicts the effective wavepacket angular uncertainty relations for $Δm Δϕ$, $Δj Δχ$, and $ΔϕΔθ$, and the position of the particle-wavepacket angular motion on the orbital plane. The particle-wavepacket rotation can be experimentally probed through continuous and non-destructive $j$-rotation measurements. We also use the $j_m (ϕ,θ,χ)$ to determine well-known asymptotic expressions for Clebsch-Gordan coefficients, Wigner d-functions, the gyromagnetic ratio of elementary particles, $g=2$, and the m-state-correlation matrix elements, $<j_3 m_3|j_{1X} j_{2X}|j_3 m_3>$. Interestingly, for low j, even down to $j=1/2$, these expressions are either exact (the last two) or excellent approximations (the first two), showing that $j_m (ϕ,θ,χ)$ gives a useful spatial description of quantum-mechanical angular momentum, and provides a smooth connection with classical angular momentum. |
| title | The Vector-Model Wavefunction: spatial description and wavepacket formation of quantum-mechanical angular momenta |
| topic | Quantum Physics |
| url | https://arxiv.org/abs/2305.11456 |