Woit · §33.1 The supersymmetric oscillator
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英文 OCR 草稿
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There is an example of a physical quantum mechanical system that has exactly the behavior of this supersymmetric oscillator. A charged particle confined to a plane, coupled to a magnetic field perpendicular to the plane, can be described by a Hamiltonian that can be put in the bosonic oscillator form (to show this, we need to know how to couple quantum systems to electromagnetic fields, which we will come to in chapter 45). The equally spaced energy levels are known as “Landau levels”. If the particle has spin , there will be an additional term in the Hamiltonian coupling the spin and the magnetic field, exactly the one we have seen in our study of the two-state system. This additional term is precisely the Hamiltonian of a fermionic oscillator. For the case of gyromagnetic ratio , the coeficients match up so that we have exactly the supersymmetric oscillator described above, with exactly the pattern of energy levels seen there.
来源与版本
正文：英文 OCR 草稿
来源版本：2025-10-20
原文位置：§33.1；PDF 页 371；印刷页 354
处理记录：MinerU OCR
核对状态：英文 OCR 草稿 · 待校对
中文机译草稿
Author-named definition; machine draft source identity and human-review status are preserved.
机器草稿中的定义提及
有一个物理量子力学系统的例子，其行为恰好与这个超对称振子相同。一个被限制在平面内并耦合于垂直于该平面的磁场的带电粒子，可以用一个能够写成玻色子振子形式的哈密顿算子来描述（为了证明这一点，我们需要知道如何将量子系统与电磁场耦合，这将在第45章中讨论）。等间距的能级被称为“朗道能级”。如果粒子具有自旋，哈密顿算子中将出现一个耦合自旋与磁场的附加项，正是我们在二态系统研究中见过的那个项。这个附加项恰好是费米子振子的哈密顿算子。对于旋磁比 的情形，系数相互匹配，因此我们恰好得到上述超对称振子，并且能级模式也恰好与那里所见相同。
来源与版本
正文：中文机译草稿
来源版本：2025-10-20
原文位置：§33.1；PDF 页 371；印刷页 354
处理记录：MinerU OCR；machine-translation/deepseek-v4.1-flash
核对状态：中文机译草稿 · 待校对