(a) How many different 6 d states does hydrogen have? (b)...

(a) How many different $6 d$ states does hydrogen have? (b) Which of the states in part (a) has the largest angle between $\vec{L}$ and the $z$ -axis, and what is that angle? (c) Which of the states in part (a) has the smallest angle between $\vec{L}$ and the $z$ -axis, and what is that angle?

(a) How many different $6 d$ states does hydrogen have?
(b) Which of the states in part (a) has the largest angle between $\vec{L}$ and the $z$ -axis, and what is that angle? (c) Which of the states in part (a) has the smallest angle between $\vec{L}$ and the $z$ -axis, and what is that angle?

Key Concepts

Orbital Quantum Number (l)

The orbital quantum number defines the magnitude of an electron's orbital angular momentum and determines the shape of its orbital. In hydrogen, d states are characterized by l = 2, which implies a fixed angular momentum magnitude of ?[2(2+1)]? = ?6?.

Magnetic Quantum Number (m)

The magnetic quantum number quantizes the projection of the orbital angular momentum along a chosen axis (typically the z-axis). For a given l value, m can take integer values ranging from -l to +l. For l = 2, this results in five possible m values: -2, -1, 0, 1, and 2.

Degeneracy of Atomic Orbitals

Degeneracy refers to the number of quantum states that have the same energy. In the context of hydrogen's d states, the degeneracy associated with the orbital angular momentum (ignoring spin) corresponds to the number of allowed m values, which is five for l = 2. If spin is considered, each spatial orbital state may be further doubled.

Angular Momentum Orientation and Quantization

The orientation of the orbital angular momentum vector is quantized relative to an external axis (usually the z-axis). The relation Lz = m? and the total magnitude |L| = ?[l(l+1)]? relate to the angle ? between the angular momentum vector and the z-axis via cos? = m/?[l(l+1)]. This formula shows that states with higher |m| correspond to smaller angles (closer alignment with the z-axis), while lower |m| lead to larger angles.

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