Chapter 41

A hydrogen atom in a 3\(p\) state is placed in a uniform external magnetic field \(\vec B\). Consider the interaction of the magnetic field with the atom's orbital magnetic dipole moment. (a) What field magnitude \(B\) is required to split the 3\(p\) state into multiple levels with an energy difference of 2.71 \(\times\) 10\(^{-5}\) eV between adjacent levels? (b) How many levels will there be?

A hydrogen atom undergoes a transition from a 2\(p\) state to the 1\(s\) ground state. In the absence of a magnetic field, the energy of the photon emitted is 122 nm. The atom is then placed in a strong magnetic field in the z-direction. Ignore spin effects; consider only the interaction of the magnetic field with the atom's orbital magnetic moment. (a) How many different photon wavelengths are observed for the 2p \(\rightarrow\) 1s transition? What are the \(m$$_l\) values for the initial and final states for the transition that leads to each photon wavelength? (b) One observed wavelength is exactly the same with the magnetic field as without. What are the initial and final \(m$$_l\) values for the transition that produces a photon of this wavelength? (c) One observed wavelength with the field is longer than the wavelength without the field. What are the initial and final \(m$$_l\) values for the transition that produces a photon of this wavelength? (d) Repeat part (c) for the wavelength that is shorter than the wavelength in the absence of the field.

A hydrogen atom in the \(n = 1\), \(ms = - {1\over2}\) state is placed in a magnetic field with a magnitude of 1.60 \(T\) in the \(+z\)- direction. (a) Find the magnetic interaction energy (in electron volts) of the electron with the field. (b) Is there any orbital magnetic dipole moment interaction for this state? Explain. Can there be an orbital magnetic dipole moment interaction for \(n\neq 1\)?

\(\textbf{Classical Electron Spin}\). (a) If you treat an electron as a classical spherical object with a radius of 1.0 \(\times\) 10\(^{-17}\) m, what angular speed is necessary to produce a spin angular momentum of magnitude \(\sqrt{3\over4}\hslash\) ? (b) Use \(v = r\omega\) and the result of part (a) to calculate the speed \(v\) of a point at the electron's equator. What does your result suggest about the validity of this model?

Calculate the energy difference between the \(m_s = {1\over2}\) ("spin up") and \(m_s = - {1\over2}\) ("spin down") levels of a hydrogen atom in the \(1s\) state when it is placed in a 1.45-T magnetic field in the \(negative\) \(z\)-direction. Which level, \(m_s = {1\over2}\) or \(ms = - {1\over2}\) , has the lower energy?

A hydrogen atom in a particular orbital angular momentum state is found to have \(j\) quantum numbers \({7\over2}\) and \({9\over2}\) . (a) What is the letter that labels the value of \(l\) for the state? (b) If \(n = 5\), what is the energy difference between the \(j = {7\over2}\) and \(j = {9\over2}\) levels?

Make a list of the four quantum numbers \(n, l, m_l\) , and \(m_s\) for each of the 10 electrons in the ground state of the neon atom. Do \(not\) refer to Table 41.2 or 41.3.

For germanium (\(Ge, Z = 32\)), make a list of the number of electrons in each subshell (\(1s, 2s, 2p,\dots\)). Use the allowed values of the quantum numbers along with the exclusion principle; do not refer to Table 41.3.

(a) Write out the ground-state electron configuration (\(1s^2, 2s^2,\dots\)) for the beryllium atom. (b) What element of nextlarger \(Z\) has chemical properties similar to those of beryllium? Give the ground-state electron configuration of this element. (c) Use the procedure of part (b) to predict what element of nextlarger \(Z\) than in (b) will have chemical properties similar to those of the element you found in part (b), and give its ground-state electron configuration.

(a) Write out the ground-state electron configuration (\(1s^2, 2s^2,\dots\)) for the carbon atom. (b) What element of nextlarger \(Z\) has chemical properties similar to those of carbon? Give the ground-state electron configuration for this element.