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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 -2魔, -魔, 0, 魔, and 2魔.

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 magnesium ion \(\mathrm{Mg}^{2+}\) is accelerated through a potential difference of \(22 \mathrm{kV}\). What is the de Broglie wavelength of this ion?
A beam of neutrons is used to study molecular structure through a series of diffraction experiments. A beam of neutrons with a wide range of de Broglie wavelengths comes from the core of a nuclear reactor. In a time-offlight technique, used to select neutrons with a small range of de Broglie wavelengths, a pulse of neutrons is allowed to escape from the reactor by opening a shutter very briefly. At a distance of \(16.4 \mathrm{m}\) downstream, a second shutter is opened very briefly 13.0 ms after the first shutter. (a) What is the speed of the neutrons selected? (b) What is the de Broglie wavelength of the neutrons? (c) If each shutter is open for 0.45 ms, estimate the range of de Broglie wavelengths selected.

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