Consider states with angular momentum quantum number I=2....

Consider states with angular momentum quantum number $I=2$. (a) In units of $\hbar$, what is the largest possible value of $L_{z}$ ? (b) In units of $\hbar,$ what is the value of $L$ ? Which is larger: $L$ or the maximum possible $L_{z}$ ? (c) For each allowed value of $L_{z},$ what angle does the vector $\vec{L}$ make with the $+z$ -axis? How does the minimum angle for $l=2$ compare to the minimum angle for $l=3$ calculated in Example 41.3 ?

Key Concepts

Angular Momentum Vector Orientation

The orientation of the angular momentum vector relative to the z-axis is determined by the relation cos(?) = m/?[l(l + 1)], where ? is the angle between the vector and the positive z-axis. This relation shows that different m values correspond to different angles, and the state with m = l gives the minimum angle between the angular momentum vector and the z-axis. Comparing different l values (such as l = 2 versus l = 3) illustrates how the minimum possible angle decreases as l increases.

Total Angular Momentum Magnitude

The magnitude of the angular momentum vector is given by the formula ?[l(l + 1)] in units of ?. This value exceeds the maximum z-component value (l) for any l > 0, reflecting the inherently three-dimensional character of angular momentum in quantum mechanics.

Relationship Between Lz and Total Angular Momentum

Because the magnitude of the angular momentum vector is ?[l(l + 1)] and its maximum projection is l, the total angular momentum is always greater than its maximum z-component. This discrepancy demonstrates that even in the state with maximum Lz, the angular momentum vector is not completely aligned with the z-axis.

Angular Momentum Quantum Numbers

In quantum mechanics, angular momentum is characterized by quantum numbers that determine its magnitude and its projection along a chosen axis (commonly the z-axis). The total angular momentum quantum number (often denoted as l or I) and the magnetic quantum number (m) define a state, with m taking on integer values between -l and +l.

Angular Momentum Projection (Lz)

The z-component of angular momentum is quantized and expressed in units of ?. Its allowed values are determined by the magnetic quantum number m. The largest possible value of Lz is achieved when m equals the total angular momentum quantum number, thereby giving Lz = l in units of ?.

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