B.10 Chapters 21 and 22

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

Problem 1:

Consider the classical Hamiltonian function for a particle moving in a central potential

h = \frac{1}{2 m} (p_{1}^{2} + p_{2}^{2} + p_{3}^{2}) + V (r)

where

r^{2} = q_{1}^{2} + q_{2}^{2} + q_{3}^{2}

• Show that the angular momentum functions $l_{j}$ satisfy

{l_{j}, h} = 0

and note that this implies that the $l_{j}$ are conserved functions along classical trajectories.

• Show that in the quantized theory the angular momentum operators and the $S O (3)$ Casimir operator satisfy

[L_{j}, H] = 0, [L^{2}, H] = 0

• Show that for a fixed energy E, the subspace $H_{E} \subset H$ of states of energy E will be a Lie algebra representation of $S O (3)$ . Decomposing into irreducibles, this can be characterized by the various spin values l that occur, together with their multiplicity.

• Show that if a state of energy $E$ lies in a spin-l irreducible representation of $S O (3)$ at time $t = 0,$ it will remain in a spin-l irreducible representation at later times.

Problem 2:

w = \frac{1}{m} (l \times p) + e^{2} \frac{q}{∣ q ∣}

is the Lenz vector, show that its components satisfy

{w_{j}, h} = 0

for the Hydrogen atom Hamiltonian $h .$

Problem 3:

For the one dimensional quantum harmonic oscillator:

• Compute the expectation values in the energy eigenstate $∣ n ⟩$ of the following operators

Q, P, Q^{2}, P^{2}

and

Q^{4}

• Use these to find the standard deviations in the statistical distributions of observed values of $q$ and $p$ in these states. These are

Δ Q = ⟨ n ∣ Q^{2} ∣ n ⟩ - ⟨ n ∣ Q ∣ n ⟩^{2}, Δ P = ⟨ n ∣ P^{2} ∣ n ⟩ - ⟨ n ∣ P ∣ n ⟩^{2}

• For two energy eigenstates $∣ n ⟩$ and $∣ n^{'} ⟩$ , find

⟨ n^{'} ∣ Q ∣ n ⟩ and ⟨ n^{'} ∣ P ∣ n ⟩

Problem 4:

Show that the functions $1, z, \overline{z}, z \overline{z}$ of section 22.4 give a basis of a Lie algebra (with Lie bracket the Poisson bracket of that section). Show that this is a semidirect product Lie algebra, and that the harmonic oscillator state space gives a representation of this Lie algebra.

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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.10 Chapters 21 and 22