17.4 Quantization and symmetries

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

The Schr¨odinger representation is thus a Lie algebra representation providing observables corresponding to elements of the Lie algebras $h_{2 d + 1}$ (linear combinations of $Q_{j}$ and $P_{k})$ and $sp (2 d, R)$ (linear combinations of degree-two combinations of $Q_{j}$ and $P_{k})$ . The observables that commute with the Hamiltonian operator $H$ will make up a Lie algebra of symmetries of the quantum system, and will take energy eigenstates to energy eigenstates of the same energy. Some examples for the physical case of $d = 3$ are:

• The group $R^{3}$ of translations in coordinate space is a subgroup of the Heisenberg group and has a Lie algebra representation as linear combinations of the operators $- i P_{j}$ . If the Hamiltonian is position-independent, for instance the free particle case of

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

then the momentum operators correspond to symmetries. Note that the position operators $Q_{j}$ do not commute with this Hamiltonian, and so do not correspond to a symmetry of the dynamics.

• The group SO(3) of spatial rotations is a subgroup of $S p (6, R)$ , with so $(3) \subset sp (6, R)$ given by the quadratic polynomials in equation 16.14 for A an antisymmetric matrix. Quantizing, the operators

- i (Q_{2} P_{3} - Q_{3} P_{2}), - i (Q_{3} P_{1} - Q_{1} P_{3}), - i (Q_{1} P_{2} - Q_{2} P_{1})

provide a basis for a Lie algebra representation of so(3). This phenomenon will be studied in detail in chapter 19.2 where we will find that for the Schr¨odinger representation on position-space wavefunctions, these are the same operators that were studied in chapter 8 under the name $ρ^{'} (l_{j})$ . They will be symmetries of rotationally invariant Hamiltonians, for instance the free particle as above, or the particle in a potential

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

when the potential only depends on the combination $Q_{1}^{2} + Q_{2}^{2} + Q_{3}^{2}$

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正文：英文 · OCR 机器稿 · 待校对

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原书 PDF · 印刷页 197、198、199、200、201、202、203

来源版本：2025-10-20

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17.4 Quantization and symmetries

17.4 Quantization and symmetries

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