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Chapter 40: Problem 15

Calculate the (a) smaller and (b) larger value of the semiclassical angle between the electron spin angular momentum vector and the magnetic field in a Stern-Gerlach experiment. Bear in mind that the orbital angular momentum of the valence electron in the silver atom is zero.

Short Answer

Expert verified
The smaller angle is \(54.7^\circ\), and the larger angle is \(125.3^\circ\).

Step by step solution

01

Understanding the Stern-Gerlach Experiment

The Stern-Gerlach experiment involves sending atoms through a non-uniform magnetic field. Silver atoms are often used, where the orbital angular momentum (\( \mathbf{L} \)) is zero, leaving only the electron spin angular momentum (\( \mathbf{S} \)) to be observed. The spin angular momentum is a quantum mechanical property that gives rise to different paths the atoms can take when passed through the magnetic field.
02

Electron Spin Angular Momentum

For an electron, the total spin angular momentum \( S \) is given by \( S = \sqrt{s(s+1)} \hbar \). For electrons, \( s = \frac{1}{2} \), so \( S = \sqrt{\frac{1}{2}(\frac{1}{2} + 1)} \hbar = \frac{\sqrt{3}}{2} \hbar \). This defines the magnitude of the angular momentum vector for the spin of the electron.
03

Zeeman Effect and Magnetic Field Interaction

In the presence of a magnetic field, the component of spin angular momentum along the field (\( S_z \)) takes discrete values \( m_s \hbar \), where \( m_s = \pm \frac{1}{2} \) for an electron. Thus, \( S_z = \pm \frac{1}{2} \hbar \). The angle \( \theta \) between the spin vector \( \mathbf{S} \) and the magnetic field direction can be found using the cosine of this angle: \( \cos \theta = \frac{S_z}{S} \).
04

Calculating the Angles

Using the known values, \( \cos \theta = \frac{\pm \frac{1}{2} \hbar}{\frac{\sqrt{3}}{2} \hbar} \), simplifying gives \( \cos \theta = \pm \frac{1}{\sqrt{3}} \). This results in two possible angles: \( \theta = \cos^{-1}\left(\frac{1}{\sqrt{3}}\right) \) for the smaller angle and \( \theta = \cos^{-1}\left(-\frac{1}{\sqrt{3}}\right) \) for the larger angle.

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Key Concepts

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

Spin Angular Momentum
Spin angular momentum is a fundamental concept in quantum mechanics. It is an intrinsic form of angular momentum carried by elementary particles, such as electrons. Unlike classical angular momentum, spin angular momentum does not arise from the particle physically spinning. Instead, it is a quantum property that is inherently part of the nature of particles. This means it cannot be directly visualized or fully compared with classical ideas.

Key features of spin angular momentum include:
  • It is quantized, meaning it can only take on specific values. For an electron, the spin quantum number is always \( s = \frac{1}{2} \), leading to a total spin angular momentum of \( S = \frac{\sqrt{3}}{2} \hbar \).
  • The projection of spin along any chosen axis, such as the z-axis in a magnetic field, also quantizes. For electrons, these projections are \( +\frac{1}{2} \hbar \) and \( -\frac{1}{2} \hbar \).
Understanding these concepts is essential for interpreting experiments like the Stern-Gerlach experiment, where the effects of spin angular momentum are directly observed.
Quantum Mechanics
Quantum mechanics is the branch of physics that deals with the behavior of particles at atomic and subatomic levels. It introduces principles that defy classical physics by describing how particles like electrons and photons behave. This behavior includes phenomena like uncertainty, quantization, and superposition.

Some fundamental principles of quantum mechanics are:
  • Particles exhibit wave-particle duality, acting as both particles and waves depending on how they are observed.
  • Physical quantities are quantized in discrete units, such as energy levels in atoms and spin angular momentum.
In the context of the Stern-Gerlach experiment, quantum mechanics explains why particles take specific discrete paths when moving through a magnetic field. This is due to the quantization of the spin, which forces particles to align along particular projections of the magnetic field. Hence, viewing phenomena like the separation of particles is fundamental to the study of quantum systems.
Zeeman Effect
The Zeeman effect is the splitting of spectral lines in the presence of a magnetic field. It occurs because the magnetic field interacts with the magnetic moments of electrons within atoms, altering their energy levels. This interaction is directly related to spin angular momentum. The Stern-Gerlach experiment demonstrates a similar concept through spatial separation due to magnetic interactions.

Key points about the Zeeman effect include:
  • When atoms are subjected to a magnetic field, the energy levels are split according to different possible orientations of angular momentum.
  • The splitting magnitude depends on the strength of the magnetic field and the magnetic properties of the particles.
The Zeeman effect provides insight into the quantum nature of electrons and their interactions with magnetic fields. It helps in measuring and understanding the behavior of atomic and subatomic particles, as well as in applications like magnetic resonance imaging (MRI) and astrophysical observations.

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Most popular questions from this chapter

Show that a moving electron cannot spontaneously change into an \(x\) -ray photon in free space. A third body (atom or nucleus) must be present. Why is it needed? (Hint: Examine the conservation of energy and momentum.)

A molybdenum \((Z=42)\) target is bombarded with \(35.0 \mathrm{keV}\) electrons and the \(x\) -ray spectrum of Fig. \(40-13\) results. The \(K_{\beta}\) and \(K_{a}\) wavelengths are \(63.0\) and \(71.0 \mathrm{pm}\), respectively. What photon energy corresponds to the (a) \(K_{\beta}\) and (b) \(K_{\alpha}\) radiation? The two radiations are to be filtered through one of the substances in the following table such that the substance absorbs the \(K_{\beta}\) line more strongly than the \(K_{\alpha}\) line. A substance will absorb radiation \(x_{1}\) more strongly than it absorbs radiation \(x_{2}\) if a photon of \(x_{1}\) has enough energy to eject a \(K\) electron from an atom of the substance but a photon of \(x_{2}\) does not. The table gives the ionization energy of the \(K\) electron in molybdenum and four other substances. Which substance in the table will serve (c) best and (d) second best as the filter? $$ \begin{array}{llllll} \hline & \mathrm{Zr} & \mathrm{Nb} & \mathrm{Mo} & \mathrm{Tc} & \mathrm{Ru} \\\ \hline Z & 40 & 40 & 42 & 43 & 44 \\ E_{K}(\mathrm{keV}) & 18.00 & 18.99 & 20.00 & 21.04 & 22.12 \end{array} $$

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 \(\vec{\mu} ?(\mathrm{c})\) What is the largest possible value of \(m_{e}\), (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 \(L\), and \(\vec{L}\) ? What is the value of angle \(\theta\) for \((\mathrm{g})\) the second largest possible value of \(m_{\ell}\) and (h) the smallest (that is, most negative) possible value of \(m_{i}\) ?

Knowing that the minimum x-ray wavelength produced by \(40.0 \mathrm{keV}\) electrons striking a target is \(31.1 \mathrm{pm}\), determine the Planck constant \(h\)

By measuring the go-and-return time for a laser pulse to travel from an Earth- bound observatory to a reflector on the Moon, it is possible to measure the separation between these bodjes. (a) What is the predicted value of this time? (b) The separation can be measured to a precision of about \(15 \mathrm{~cm}\). To what uncertainty in travel time does this correspond? (c) If the laser beam forms a spot on the Moon \(3 \mathrm{~km}\) in diameter, what is the angular divergence of the beam? x
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