advanced_tools:group_theory:su2

This shows you the differences between two versions of the page.

Both sides previous revision Previous revision Next revision | Previous revision | ||

advanced_tools:group_theory:su2 [2020/12/05 18:29] edi [Intuitive] |
advanced_tools:group_theory:su2 [2020/12/26 22:52] edi [Intuitive] |
||
---|---|---|---|

Line 3: | Line 3: | ||

<tabbox Intuitive> | <tabbox Intuitive> | ||

- | The Lie group $SU(2)$ describes all possible rotations of a spinorial object, that is, an object that needs to be rotated 720 degrees before returning to its initial state. A good example for such an object is a cube that is attached to a wall by belts: see the animations here [[https://en.wikipedia.org/wiki/Spinor]]. In physics, an important spinorial object is the fermion (e.g., an electron). | + | The Lie group $SU(2)$ describes all possible 3D rotations of a spinorial object, that is, an object that needs to be rotated 720 degrees before returning to its initial state. A good example for such an object is a cube that is attached to a wall by belts: see the animations here [[https://en.wikipedia.org/wiki/Spinor]]. In physics, an important spinorial object is the fermion (e.g., an electron). |

For small rotations $SU(2)$ is identical to $SO(3)$, that is, both groups have the same Lie algebra. | For small rotations $SU(2)$ is identical to $SO(3)$, that is, both groups have the same Lie algebra. | ||

Line 27: | Line 27: | ||

**Representations** | **Representations** | ||

- | The diagram below shows the defining 2-dimensional representation of $SU(2)$, which describes spin-$1/2$ particles, in its upper branch. A 3-dimensional representations of the same group, which describes spin-$1$ particles, is shown in the lower branch. For an explanation of this diagram and more representations of $SU(2)$ see [[https://esackinger.wordpress.com/|Fun with Symmetry]]. | + | The diagram below shows the defining (2-dimensional) representation of $SU(2)$ in its upper branch and a 3-dimensional representations of the same group in the lower branch. An important application of these two representations is the rotation of the quantum state of a spin-1/2 and a spin-1 particle, respectively. For a more detailed explanation of this diagram and more representations of $SU(2)$ see [[https://esackinger.wordpress.com/|Fun with Symmetry]]. |

[{{ :advanced_tools:group_theory:su2_qm_spin.jpg?nolink }}] | [{{ :advanced_tools:group_theory:su2_qm_spin.jpg?nolink }}] |

advanced_tools/group_theory/su2.txt ยท Last modified: 2020/12/26 22:52 by edi

Except where otherwise noted, content on this wiki is licensed under the following license: CC Attribution-Share Alike 4.0 International