**Problem 7: Quantum States of Singly-Ionized Helium***Ignoring the fine structure splitting, hyperfine structure splitting, etc., such that energy levels depend only on the principal quantum number \( n \), how many distinct states of singly-ionized helium (\( Z = 2 \)) have energy \( E = -13.6 \, \text{eV} \)? Write out all the quantum numbers \( (n, \ell, m_\ell, m_s) \) describing each distinct state. (Recall that the ground state energy of hydrogen is \( E_1 = -13.6 \, \text{eV} \), and singly-ionized helium may be treated as a hydrogen-like atom.)***Explanation:**This problem involves finding the distinct quantum states for singly-ionized helium where the energy level is described only by the principal quantum number \( n \). It assumes a simplified model of atom where other effects such as fine structure are ignored.1. **Quantum Numbers**: - \( n \): Principal quantum number. - \( \ell \): Orbital angular momentum quantum number, which can take values from \( 0 \) to \( n-1 \). - \( m_\ell \): Magnetic quantum number, with values from \( -\ell \) to \( \ell \). - \( m_s \): Spin quantum number, which can be \( +\frac{1}{2} \) or \( -\frac{1}{2} \).2. **Energy Level Consideration**: - Since \( Z = 2 \) for helium, and the problem mentions \( E = -13.6 \, \text{eV} \), this corresponds to the \( n=2 \) energy level, because for hydrogen-like atoms, \( E_n = -13.6 \, \text{eV} \times \frac{Z^2}{n^2} \). - For \( Z = 2 \), \( E_1 = -54.4 \, \text{eV} \). For \( n=2 \), \( E_2 = -13.6 \, \text{eV} \).3. **Quantum States Enumeration**: - For \( n=2 \): \( \ell \) can be \( 0 \) or \( 1 \). - If

Energy levels of a singly ionized He atom(Z=2) is given by En=-Z2n2×13.6=-22n2×13.6 eV Hence from…

Ignoring the fine structure splitting, hyperfine structure splitting, etc., such that energy levels depend only on the principal quantum number n, how many distinct states of singly-ionized helium (Z = 2) have energy E= -13.6 eV? Write out all the quantum numbers (n, l, m, m,) describing each distinct state. (Recall that the ground state energy of hydrogen is E₁ = -13.6 eV, and singly-ionized helium may be treated as a hydrogen-like atom.)

Ignoring the fine structure splitting, hyperfine structure splitting, etc., such that energy levels depend only on the principal quantum number n, how many distinct states of singly-ionized helium (Z = 2) have energy E= -13.6 eV? Write out all the quantum numbers (n, l, m, m,) describing each distinct state. (Recall that the ground state energy of hydrogen is E₁ = -13.6 eV, and singly-ionized helium may be treated as a hydrogen-like atom.)

College Physics

11th Edition

ISBN:9781305952300

Author:Raymond A. Serway, Chris Vuille

Publisher:Raymond A. Serway, Chris Vuille

Chapter1: Units, Trigonometry. And Vectors

Section: Chapter Questions

Problem 1CQ: Estimate the order of magnitude of the length, in meters, of each of the following; (a) a mouse, (b)...

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