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Chapter 40: Problem 49

An electron is in a state with \(\ell=3 .\) (a) What multiple of \(\hbar\) gives the magnitude of \(\vec{L}\) ? (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 \(\mu_{\mathrm{orb}, z}\) ? (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 \(m_{\ell}\) and \((\mathrm{h})\) the smallest (that is, most negative) possible value of \(m_{\ell}\) ?

Short Answer

Expert verified
(a) 2\sqrt{3} (b) -2\sqrt{3} (c) 3; (d) 3; (e) -3; (f) 30掳; (g) 54.7掳; (h) 150掳.

Step by step solution

01

Determine magnitude of \(\vec{L}\)

The magnitude of the orbital angular momentum \(\vec{L}\) is given by \(L = \sqrt{\ell(\ell+1)} \hbar\). For \(\ell=3\), we calculate \(L = \sqrt{3(3+1)} \hbar = \sqrt{12} \hbar = 2\sqrt{3} \hbar\). Thus, the magnitude of \(\vec{L}\) is \(2\sqrt{3}\) times \(\hbar\).
02

Determine magnitude of \(\vec{\mu}\)

The magnetic dipole moment \(\mu\) for an electron is given by the formula: \(\mu = -\frac{e}{2m_e}L\) where \(e\) is the charge of the electron and \(m_e\) is the electron mass. We use \(\mu_B = \frac{e\hbar}{2m_e}\) (Bohr magneton) to express \(\mu\) in terms of \(\mu_B\). Therefore, \(\mu = -\frac{L}{\hbar} \mu_B = -\sqrt{\ell(\ell+1)} \mu_B\). For \(\ell = 3\), \(\mu = -\sqrt{12} \mu_B = -2\sqrt{3} \mu_B\).
03

Determine largest possible value of \(m_\ell\)

The magnetic quantum number \(m_\ell\) can take values in the range \(-\ell, \ldots, \ell\). For \(\ell = 3\), the largest \(m_\ell\) is 3.
04

Determine multiple of \(\hbar\) for \(L_z\), when \(m_\ell\) is largest

For \(m_\ell = 3\), the z-component of angular momentum is \(L_z = m_\ell \hbar = 3 \hbar\).
05

Determine multiple of \(\mu_B\) for \(\mu_{\text{orb}, z}\), when \(m_\ell\) is largest

The orbital magnetic moment component along z is \(\mu_{\text{orb}, z} = -m_\ell \mu_B = -3 \mu_B\).
06

Determine semiclassical angle \(\theta\) for largest \(m_\ell\)

The semiclassical angle \(\theta\) between \(L_z\) and \(\vec{L}\) is given by \(\cos\theta = \frac{L_z}{L}\). For \(m_\ell = 3\), \(\cos\theta = \frac{3 \hbar}{2\sqrt{3} \hbar} = \frac{3}{2\sqrt{3}} = \frac{\sqrt{3}}{2}\). Thus, \(\theta = 30^\circ\).
07

Determine \(\theta\) for second largest \(m_\ell\)

For the second largest \(m_\ell = 2\), \(L_z = 2\hbar\). Then, \(\cos\theta = \frac{2 \hbar}{2\sqrt{3} \hbar} = \frac{1}{\sqrt{3}}\). Thus, \(\theta = 54.7^\circ\).
08

Determine \(\theta\) for smallest possible \(m_\ell\)

For \(m_\ell = -3\), \(L_z = -3\hbar\). Then, \(\cos\theta = \frac{-3 \hbar}{2\sqrt{3} \hbar} = -\frac{\sqrt{3}}{2}\). Thus, \(\theta = 150^\circ\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Angular Momentum
In quantum mechanics, angular momentum plays a pivotal role in describing the rotational motion of particles. It is a fundamental property, just like energy and momentum, but specifically pertains to systems with circular symmetry.

Angular momentum in quantum mechanics is quantized, meaning it can only take on certain discrete values. The magnitude of the orbital angular momentum, denoted as \(\vec{L}\), is determined by the equation:
  • \( L = \sqrt{\ell(\ell+1)} \hbar \)
Here, \(\ell\) is the azimuthal quantum number or angular momentum quantum number, and \(\hbar\) is the reduced Planck's constant. This quantization reflects the wave-like nature of particles at microscopic scales.

For different values of \(\ell\), angular momentum can assume various magnitudes, influencing the spatial orientation of the quantum state. Understanding these magnitudes allows one to predict how electrons are distributed around the nucleus in an atom, influencing both chemical bonding and spectroscopy.
Magnetic Quantum Number
The magnetic quantum number, represented as \(m_\ell\), is another key aspect of quantum mechanics. This number provides information about the orientation of the angular momentum vector \(\vec{L}\) in space.

The values \(m_\ell\) can assume range from \(-\ell\) to \(\ell\) in integer steps. Each value of \(m_\ell\) represents a different possible orientation of the electron's angular momentum vector around the nucleus.
  • For an angular quantum number \(\ell = 3\), \(m_\ell\) can take on any of the following values: -3, -2, -1, 0, 1, 2, 3.
The significance of the magnetic quantum number lies in its connection to the z-component of the angular momentum (\(L_z = m_\ell \hbar\)), determining the component of the angular momentum in a given direction. This is especially useful in magnetic fields, where the electron's magnetic properties become relevant.
Bohr Magneton
The Bohr magneton \(\mu_B\) is a fundamental physical constant in quantum mechanics that describes the magnetic moment of an electron due to its angular momentum.

It is the natural unit for expressing the magnetic moment of electrons, defined as:
  • \( \mu_B = \frac{e\hbar}{2m_e} \)
where \(e\) is the elementary charge and \(m_e\) is the electron mass.

The Bohr magneton provides a useful scale for describing the magnetic moments associated with the electron's orbital angular momentum. For example, the magnetic dipole moment \(\mu\) associated with an electron in orbit is typically expressed as a multiple of \(\mu_B\).

In practical applications, these values inform the study of electron behaviors in magnetic fields, allowing scientists to predict and manipulate atomic and molecular magnetic properties, which underpin technologies like MRI and other magnetic resonance techniques.

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Most popular questions from this chapter

Suppose that a hydrogen atom in its ground state moves \(80 \mathrm{~cm}\) through and perpendicular to a vertical magnetic field that has a magnetic field gradient \(d B / d z=1.6 \times 10^{2} \mathrm{~T} / \mathrm{m}\). (a) What is the magnitude of force exerted by the field gradient on the atom due to the magnetic moment of the atom's electron, which we take to be 1 Bohr magneton? (b) What is the vertical displacement of the atom in the \(80 \mathrm{~cm}\) of travel if its speed is \(2.5 \times 10^{5} \mathrm{~m} / \mathrm{s}\) ?

Through what minimum potential difference must an electron in an \(\mathrm{x}\)-ray tube be accelerated so that it can produce \(\mathrm{x}\) rays with a wavelength of \(0.100 \mathrm{~nm}\) ?

A recently named element is darmstadtium (Ds), which has 110 electrons. Assume that you can put the 110 electrons into the atomic shells one by one and can neglect any electronelectron interaction. With the atom in ground state, what is the spectroscopic notation for the quantum number \(\ell\) for the last electron?

The active volume of a laser constructed of the semiconductor \(\mathrm{GaAlAs}\) is only \(200 \mu \mathrm{m}^{3}\) (smaller than a grain of sand), and yet the laser can continuously deliver \(5.0 \mathrm{~mW}\) of power at a wavelength of \(0.80 \mu \mathrm{m}\). At what rate does it generate photons?

A hypothetical atom has energy levels uniformly separated by \(1.2 \mathrm{eV}\). At a temperature of \(2000 \mathrm{~K}\), what is the ratio of the number of atoms in the 13 th excited state to the number in the 11 th excited state?
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