The vector model (Problem 69 ) gives some insight into the...

The vector model (Problem 69 ) gives some insight into the uncertainty principle for angular momentum, which is $\Delta L_{z} \Delta \phi \geq \hbar$ for the $z$ component. Here $\phi$ is the angular position measured in the plane perpendicular to the $z$ axis. Once $m_{\ell}$ for an atom is known, $L_{z}$ is known precisely, so $\Delta L_{z}=0$. (a) What does this tell us about $\phi ?(b)$ What can you say about $L_{x}$ and $L_{y},$ which are not quantized (only $L$ and $L_{z}$ are $) ?$ (c) Show that although $L_{x}$ and $L_{y}$ are not quantized, nonetheless $\left(L_{x}^{2}+L_{y}^{2}\right)^{1 / 2}=\left[\ell(\ell+1)-m_{\ell}^{2}\right]^{1 / 2} \hbar$ is.

The vector model (Problem 69 ) gives some insight into the uncertainty principle for angular momentum, which is $\Delta L_{z} \Delta \phi \geq \hbar$
for the $z$ component. Here $\phi$ is the angular position measured in the plane perpendicular to the $z$ axis. Once $m_{\ell}$ for an atom is known, $L_{z}$ is known precisely, so $\Delta L_{z}=0$. (a) What does this tell us about $\phi ?(b)$ What can you say about $L_{x}$ and $L_{y},$ which are not quantized (only $L$ and $L_{z}$ are $) ?$ (c) Show that although $L_{x}$ and $L_{y}$ are not quantized, nonetheless $\left(L_{x}^{2}+L_{y}^{2}\right)^{1 / 2}=\left[\ell(\ell+1)-m_{\ell}^{2}\right]^{1 / 2} \hbar$ is.

Key Concepts

Commutation Relations of Angular Momentum Components

The non-commutative nature of the angular momentum operators, a direct consequence of their underlying algebraic structure, implies that measurements of different components disturb each other. This is reflected in the commutation relations, which mathematically encode the fact that knowing one component exactly (like Lz) makes the measurement of the other components (Lx and Ly) inherently uncertain, consistent with the uncertainty principle.

Vector Model of Angular Momentum

The vector model provides a geometric interpretation of angular momentum by treating it as a vector with a fixed magnitude and components along different axes. This model helps visualize how, even though the total angular momentum and one component (typically Lz) are quantized, the other components (Lx and Ly) remain indeterminate. However, the combination of these components, such as the magnitude of the projection in the xy-plane, is still governed by quantized rules.

Uncertainty Principle in Angular Momentum

The uncertainty principle asserts that certain pairs of observables cannot be simultaneously known to arbitrary precision. For angular momentum, this is exemplified by the relation between the measurement of a component, like Lz, and the conjugate variable, the angular position ?. When one of these quantities is known exactly, the uncertainty in the other becomes infinite, reflecting the fundamental limits imposed by quantum mechanics.

Angular Momentum Quantization

In quantum mechanics, angular momentum does not take on a continuum of values but is instead quantized. This means that both the total angular momentum and one of its components (usually Lz) can only have discrete values. The quantization arises from the intrinsic properties of wave functions and the boundary conditions imposed on them, leading to specific allowed eigenstates.

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