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Chapter 28: Problem 37

What are the possible values of \(L_{z}\) (the component of angular momentum along the \(z\) -axis) for the electron in the second excited state \((n=3)\) of the hydrogen atom?

Short Answer

Expert verified
Answer: The possible values of the electron's angular momentum component along the z-axis in the second excited state of the hydrogen atom are -2ħ, -ħ, 0, ħ, and 2ħ.

Step by step solution

01

Determine values of 'l'

Recall that the angular momentum quantum number (\(l\)) can have integer values ranging from 0 to (n-1) where n is the principal quantum number. Since \(n=3\), the possible values of \(l\) are 0, 1, and 2.
02

Determine values of 'm_l' for each value of 'l'

Recall that the magnetic quantum number (\(m_l\)) can have integer values ranging from -\(l\) to \(l\). Let's find the possible values of \(m_l\) for each value of \(l\): - When \(l=0\), the possible value of \(m_l\) is 0, as there is only one component along the z-axis. - When \(l=1\), the possible values of \(m_l\) are -1, 0, and 1. These represent the three magnetic substates for the given value of the angular momentum quantum number. - When \(l=2\), the possible values of \(m_l\) are -2, -1, 0, 1, and 2. These represent the five magnetic substates for the given value of the angular momentum quantum number.
03

Determine the possible values of \(L_z\)

Finally, we can use the formula \(L_z = m_l \hbar\), to calculate the possible values of \(L_z\) for each value of \(m_l\). We can eliminate the constant \(\hbar\) in this equation, as we are only interested in the relative ratios, and \(\hbar\) is just a scaling factor. The possible values of \(m_l\) are: -2, -1, 0, 1, and 2, which are the possible values of \(L_z\) when scaled by \(\hbar\). Thus, the electron in the second excited state (\(n=3\)) of the hydrogen atom has possible components of angular momentum along the \(z\)-axis as: \(L_z = \{-2 \hbar, -\hbar, 0, \hbar, 2 \hbar\}\)

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Most popular questions from this chapter

The particle in a box model is often used to make rough estimates of ground- state energies. Suppose that you have a neutron confined to a one-dimensional box of length equal to a nuclear diameter (say \(10^{-14} \mathrm{m}\) ). What is the ground-state energy of the confined neutron?
The particle in a box model is often used to make rough estimates of energy level spacings. Suppose that you have a proton confined to a one-dimensional box of length equal to a nuclear diameter (about \(10^{-14} \mathrm{m}\) ). (a) What is the energy difference between the first excited state and the ground state of this proton in the box? (b) If this energy is emitted as a photon as the excited proton falls back to the ground state, what is the wavelength and frequency of the electromagnetic wave emitted? In what part of the spectrum does it lie? (c) Sketch the wave function \(\psi\) as a function of position for the proton in this box for the ground state and each of the first three excited states.
Neutron diffraction by a crystal can be used to make a velocity selector for neutrons. Suppose the spacing between the relevant planes in the crystal is \(d=0.20 \mathrm{nm}\) A beam of neutrons is incident at an angle \(\theta=10.0^{\circ}\) with respect to the planes. The incident neutrons have speeds ranging from 0 to \(2.0 \times 10^{4} \quad \mathrm{m} / \mathrm{s} .\) (a) What wavelength(s) are strongly reflected from these planes? [Hint: Bragg's law, Eq. \((25-15),\) applies to neutron diffraction as well as to x-ray diffraction.] (b) For each of the wavelength(s), at what angle with respect to the incident beam do those neutrons emerge from the crystal?
The distance between atoms in a crystal of \(\mathrm{NaCl}\) is $0.28 \mathrm{nm} .$ The crystal is being studied in a neutron diffraction experiment. At what speed must the neutrons be moving so that their de Broglie wavelength is \(0.28 \mathrm{nm} ?\)
An x-ray diffraction experiment using 16 -keV x-rays is repeated using electrons instead of \(x\) -rays. What should the kinetic energy of the electrons be in order to produce the same diffraction pattern as the x-rays (using the same crystal)?
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