(a) Starting with the canonical commutation relations for...

(a) Starting with the canonical commutation relations for position and momentum (Equation 4.10), work out the following commutators: $\left[L_{z}, x\right]=i \hbar y, \quad\left[L_{z}, y\right]=-i \hbar x, \quad\left[L_{z}, z\right]=0$, $\left[L_{z}, p_{x}\right]=i \hbar p_{y}, \quad\left[L_{z}, p_{y}\right]=-i \hbar p_{x}, \quad\left[L_{z}, p_{z}\right]=0$ (b) Use these results to obtain $\left[L_{z}, L_{x}\right]=i \hbar L_{y}$ directly from Equation $4.96$. (c) Evaluate the commutators $\left[L_{z}, r^{2}\right]$ and $\left[L_{z}, p^{2}\right]$ (where, of course, $r^{2}=$ $x^{2}+y^{2}+z^{2}$ and $\left.p^{2}=p_{x}^{2}+p_{y}^{2}+p_{z}^{2}\right)$ (d) Show that the Hamiltonian $H=\left(p^{2} / 2 m\right)+V$ commutes with all three components of $\mathbf{L}$, provided that $V$ depends only on $r$. (Thus $H, L^{2}$, and $L_{z}$ are mutually compatible observables.)

(a) Starting with the canonical commutation relations for position and momentum (Equation 4.10), work out the following commutators:
$\left[L_{z}, x\right]=i \hbar y, \quad\left[L_{z}, y\right]=-i \hbar x, \quad\left[L_{z}, z\right]=0$,
$\left[L_{z}, p_{x}\right]=i \hbar p_{y}, \quad\left[L_{z}, p_{y}\right]=-i \hbar p_{x}, \quad\left[L_{z}, p_{z}\right]=0$
(b) Use these results to obtain $\left[L_{z}, L_{x}\right]=i \hbar L_{y}$ directly from Equation $4.96$.
(c) Evaluate the commutators $\left[L_{z}, r^{2}\right]$ and $\left[L_{z}, p^{2}\right]$ (where, of course, $r^{2}=$ $x^{2}+y^{2}+z^{2}$ and $\left.p^{2}=p_{x}^{2}+p_{y}^{2}+p_{z}^{2}\right)$
(d) Show that the Hamiltonian $H=\left(p^{2} / 2 m\right)+V$ commutes with all three components of $\mathbf{L}$, provided that $V$ depends only on $r$. (Thus $H, L^{2}$, and $L_{z}$ are mutually compatible observables.)

Key Concepts

Canonical Commutation Relations

These are the fundamental relationships between pairs of quantum operators, such as position and momentum, typically given by [x, p] = i?. They form the basis of quantum mechanics by encoding the uncertainty principle and establishing a non-classical algebraic structure that underpins the behavior and measurement limitations of quantum observables.

Angular Momentum Operators

Angular momentum operators in quantum mechanics are defined through the cross-product of position and momentum (for example, L = r × p). They are central to understanding rotational symmetries and dynamics in quantum systems. Their commutation relations, such as [Lz, Lx] = i?Ly, reveal the algebraic structure (Lie algebra) that governs how different components of angular momentum transform under rotations.

Operator Commutators

Commutators are mathematical expressions used to quantify the non-commutativity of operators, defined as [A, B] = AB - BA. In quantum mechanics, commutators determine how observables are related and influence the possibility of simultaneously measuring quantities. They provide insight into the underlying symmetries and conservation laws of the system.

Spherical Symmetry and Compatible Observables

Spherical symmetry refers to systems where the potential depends only on the distance from a central point (V = V(r)), which results in conservation of angular momentum. In such systems, the Hamiltonian commutes with the angular momentum operators, meaning that they share a common set of eigenstates and can be simultaneously measured. This compatibility is crucial for simplifying the analysis of quantum mechanical systems in central potentials.

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