An electron is in a state with ℓ=3. (a) What multiple of h gives the magnitude of L ? (b) What m

step 1 In this problem on the topic of atoms, we're told that an electron is in a state of l is equal t

An electron is in a state with ℓ=3. (a) What multiple of h...

An electron is in a state with $\ell=3$. (a) What multiple of $h$ gives the magnitude of $L ?$ (b) What multiple of $\mu_{\mathrm{B}}$ gives the magnitude of $\vec{\mu} ?$ (c) What is the largest possible value of $m_{\ell},$ (d) what multiple of $\hbar$ gives the corresponding value of $L_{z}$, and (e) what multiple of $\mu_{\mathrm{B}}$ gives the corresponding value of $\mu_{\text {orb }, z} ?$ (f) What is the value of the semiclassical angle $\theta$ between the directions of $L_{z}$ and $\vec{L} ?$ What is the value of angle $\theta$ for $(\mathrm{g})$ the second largest possible value of $m_{e}$ and $(\mathrm{h})$ the smallest (that is, most negative) possible value of $m_{\ell} ?$

Key Concepts

Orbital Angular Momentum Quantization

In quantum mechanics, the magnitude of the orbital angular momentum is quantized. It is given by L = ?(l(l+1)) ?, where l is the orbital quantum number. This quantization arises naturally from solving the Schrödinger equation in a central potential and directly reflects the discrete energy and angular momentum levels available to electrons in atoms.

Angular Momentum Component Quantization

The z?component of the orbital angular momentum is also quantized. It can only take on values of Lz = m? ?, where m? is the magnetic quantum number (an integer ranging from -l to +l). This quantization is a consequence of the rotational symmetry of the system and the requirement that the angular part of the wavefunction be single-valued.

Quantum Numbers

Quantum numbers, such as the orbital quantum number l and the magnetic quantum number m?, characterize the state of an electron in an atom. The quantum number l defines the magnitude of the orbital angular momentum, while m? specifies the projection of this angular momentum along a chosen axis (typically the z-axis). These numbers are fundamental in describing allowed states and transitions in quantum systems.

Magnetic Moment from Orbital Motion

The orbital motion of an electron gives rise to a magnetic moment. This magnetic moment is directly related to the orbital angular momentum by the relation ? = - (e/2m?)L (often expressed in multiples of the Bohr magneton ?B). The connection between the angular momentum and the magnetic moment is a cornerstone of understanding phenomena such as the Zeeman effect, where energy levels split in the presence of a magnetic field.

Semiclassical Interpretation of Angular Momentum

While angular momentum in quantum mechanics is inherently quantized, its vector properties can be discussed in a semiclassical context. The angle between the total angular momentum vector L and its z-component Lz is given by the relationship cos ? = Lz / |L|. This semiclassical picture provides an intuitive understanding of how the quantized levels of m? affect the orientation of L in space, despite the fact that precise direction of L is not defined in quantum mechanics.

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