B.12 Chapters 24 to 26

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

Problem 1:

Consider the harmonic oscillator in two dimensions, with the Hamiltonian

H = \frac{1}{2 m} (P_{1}^{2} + P_{2}^{2}) + \frac{1}{2} m ω^{2} (Q_{1}^{2} + Q_{2}^{2})

There are two diferent $U (1) = S O (2)$ groups acting on the phase space of this system as symmetries, with corresponding operators:

• The rotation action on position space, with a simultaneous rotation action on momentum space. The operator here will be the $d = 2$ angular momentum operator $Q_{1} P_{2} - Q_{2} P_{1}$

• Simultaneous rotations in the $q_{1}, p_{1}$ and $q_{2}, p_{2}$ planes. The operator here will be the Hamiltonian.

For each case, the state space $H = F_{2}$ will be a representation of the group $U (1) = S O (2)$ . For each energy eigenspace, which irreducible representations (weights) occur? What are the corresponding joint eigenfunctions of the two operators?

Problem 2:

Consider the harmonic oscillator in three dimensions, with the Hamiltonian

H = \frac{1}{2 m} (P_{1}^{2} + P_{2}^{2} + P_{3}^{2}) + \frac{1}{2} m ω^{2} (Q_{1}^{2} + Q_{2}^{2} + Q_{3}^{2})

• The group $S O (3)$ acts on the system by rotations of the position space $R^{3}$ and the corresponding Lie algebra action on the state space $F_{3}$ is given in section 25.4.2 as the operators

U_{l_{1}}^{'}, U_{l_{2}}^{'}, U_{l_{3}}^{'}

Exponentiating to get an $S O (3)$ representation by operators $U (g)$ , show that acting by such operators on the $a_{j}$ by conjugation

a_{j} \to U (g) a_{j} U (g)^{- 1}

one gets the same action as the standard action of a rotation on coordinates on $R^{3}$

• The energy eigenspaces are the subspaces $H^{n} \subset H$ with total number eigenvalue n. These are irreducible representations of $S U (3)$ . They are also representations of the $S O (3)$ rotation action. Derive the rule for which irreducibles of $S O (3)$ will occur in $H^{n}$

Problem 3:

Prove the relation of equation 26.16.

Problem 4:

Compute

_{τ} ⟨ 0∣ N ∣0 ⟩_{τ}

as a function of $τ,$ for $∣0 ⟩_{γ}$ the squeezed state of equation 26.19 and $N$ the usual number operator.

来源与版本

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

来源 PDF SHA-256：5a1941b2443b54d5db3d055f1e5ba390429b7a728475258017aaac87ee85a837

OCR 来源 SHA-256：7f317d1896fa748af4cb9570310ed7801a8bef7ef97dd5d1d2ea35f0099d5952

OCR 产物 SHA-256：7f317d1896fa748af4cb9570310ed7801a8bef7ef97dd5d1d2ea35f0099d5952

数学物理阅读

书架

B.12 Chapters 24 to 26