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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

A beam of neutrons has the same de Broglie wavelength as a beam of photons. Is it possible that the energy of each photon is equal to the kinetic energy of each neutron? If so, at what de Broglie wavelength(s) does this occur? [Hint: For the neutron, use the relativistic energy-momentum relation \(\left.E^{2}=E_{0}^{2}+(p c)^{2} .\right]\)
An 81 -kg student who has just studied matter waves is concerned that he may be diffracted as he walks through a doorway that is \(81 \mathrm{cm}\) across and \(12 \mathrm{cm}\) thick. (a) If the wavelength of the student must be about the same size as the doorway to exhibit diffraction, what is the fastest the student can walk through the doorway to exhibit diffraction? (b) At this speed, how long would it take the student to walk through the doorway?
A fly with a mass of \(1.0 \times 10^{-4} \mathrm{kg}\) crawls across a table at a speed of \(2 \mathrm{mm} / \mathrm{s} .\) Compute the de Broglie wavelength of the fly and compare it with the size of a proton (about $\left.1 \mathrm{fm}, 1 \mathrm{fm}=10^{-15} \mathrm{m}\right)$.
What is the ratio of the wavelength of a 0.100-keV photon to the wavelength of a 0.100-keV electron?
An electron is confined to a one-dimensional box of length \(L\) (a) Sketch the wave function for the third excited state. (b) What is the energy of the third excited state? (c) The potential energy can't really be infinite outside of the box. Suppose that \(U(x)=+U_{0}\) outside the box, where \(U_{0}\) is large but finite. Sketch the wave function for the third excited state of the electron in the finite box. (d) Is the energy of the third excited state for the finite box less than, greater than, or equal to the value calculated in part (b)? Explain your reasoning. [Hint: Compare the wavelengths inside the box.] (e) Give a rough estimate of the number of bound states for the electron in the finite box in terms of \(L\) and \(U_{0}\).
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