B.9 Chapters 18 and 19

Concept links · terms present in this machine draft; source roles are unverified: Lie algebra · Lie algebra representation · Lie bracket

Problem 1:

Starting with the Lie algebra so(3), with basis $l_{1}, l_{2}, l_{3}$ , consider new basis elements given by leaving $l_{3}$ alone and rescaling

l_{1} \to l_{1} / R, l_{2} \to - l_{2} / R

where $R$ is a real parameter. Show that in the limit $R \to \infty$ these new basis elements satisfy the Lie bracket relations for the Lie algebra of $E (2)$ . Because of this, the group $E (2)$ is sometimes said to be a “contraction” of $S O (3)$

Problem 2:

For the case of the group $E (2)$ , show that in any representation $π^{'}$ of its Lie algebra, there is a Casimir operator

∣ P ∣^{2} = π^{'} (p_{1}) π^{'} (p_{1}) + π^{'} (p_{2}) π^{'} (p_{2})

that commutes with all the Lie algebra representation operators $(i.e.$ , with $π^{'} (p_{1}), π^{'} (p_{2}), π^{'} (l))$

For the case of the group $E (3)$ , similarly show that there are two Casimir operators.

∣ P ∣^{2} = π^{'} (p_{1}) π^{'} (p_{1}) + π^{'} (p_{2}) π^{'} (p_{2}) + π^{'} (p_{3}) π^{'} (p_{3})

and

L \cdot P = π^{'} (l_{1}) π^{'} (p_{1}) + π^{'} (l_{2}) π^{'} (p_{2}) + π^{'} (l_{3}) π^{'} (p_{3})

that commute with all the Lie algebra representation operators.

Problem 3:

Show that the $E (3)$ Casimir operator $L \cdot P$ acts trivially on the $E (3)$ representation on free-particle wavefunctions of energy $E > 0$

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正文：英文 · OCR 机器稿 · 待校对

核对状态：OCR 机器稿 · 待校对

原书 PDF · 印刷页 537、538、539、540、541、542、543、544、545、546、547、548、549、550、551、552、553、554、555、556

来源版本：2025-10-20

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书架

B.9 Chapters 18 and 19