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Chapter 11: Problem 80

For hydrogen-like ions with \(Z>1\), does the degeneracy of the shell vary with \(Z ?\) Why or why not?

Short Answer

Expert verified
The degeneracy does not vary with \( Z \) as it only depends on the principal quantum number \( n \).

Step by step solution

01

Understanding Degeneracy

First, we need to understand what degeneracy means in the context of hydrogen-like ions. Degeneracy refers to the number of different states that have the same energy level. For example, in hydrogen (or hydrogen-like ions), the degeneracy of an energy level is determined by the principal quantum number, which is given by the formula: \[ g_n = n^2 \] where \( g_n \) is the degeneracy and \( n \) is the principal quantum number.
02

Principal Quantum Number

For hydrogen-like ions, the principal quantum number \( n \) is the same as in the hydrogen atom. The expression for degeneracy, \( n^2 \), remains valid irrespective of the atomic number \( Z \). This is because degeneracy is solely dependent on the principal quantum number \( n \) and not on \( Z \).
03

Dependence on Atomic Number \( Z \)

The energy levels themselves depend on the atomic number \( Z \), as the energy is given by the formula: \[ E_n = - rac{Z^2 imes 13.6 ext{ eV}}{n^2} \] However, the number of degenerate states \( g_n = n^2 \) does not change with \( Z \). This means that while the energy levels become more negative (i.e., deeper) as \( Z \) increases, the degeneracy remains constant for each principal quantum number.
04

Conclusion

For hydrogen-like ions with \( Z > 1 \), the degeneracy of the shell does not vary with \( Z \). This is because the degeneracy depends only on the principal quantum number \( n \), which remains fixed regardless of the value of \( Z \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding Degeneracy
Degeneracy is a core concept when discussing atomic structure, especially in hydrogen-like ions. Simply put, degeneracy refers to the number of different quantum states that share the same energy level in an atom.
For any principal quantum number \( n \), the degeneracy is given by the formula \( g_n = n^2 \). This means, for each energy level characterized by \( n \), there are \( n^2 \) states that have the same energy.
The term 'hydrogen-like ions' refers to ions that have only one electron, similar to hydrogen. Although they possess only one electron, different from hydrogen is their atomic number \( Z \), which is greater than one. However, the number of states, or degeneracy, for a given energy level, remains determined solely by \( n \), not by \( Z \). Thus, even as \( Z \) changes, the degeneracy value for a particular \( n \) remains constant.
Principal Quantum Number
The principal quantum number, denoted by \( n \), is one of the four quantum numbers used in quantum mechanics to describe the unique quantum state of an electron in an atom.
This number essentially indicates the main energy level or shell occupied by the electron. Higher values of \( n \) correspond to higher energy levels and, generally speaking, larger atomic orbitals.
For hydrogen or hydrogen-like ions, the principal quantum number defines the energy levels and directly influences the degeneracy, which is calculated by \( n^2 \). Importantly, \( n \) is independent of the atomic number \( Z \), meaning that degeneracy remains unaffected by changes in \( Z \).
  • The value of \( n \) begins at 1 and can theoretically go to infinity, but in practice, it is limited by the electron鈥檚 binding energies and the nature of the ions.
Atomic Number and its Role
The atomic number \( Z \) is a crucial component of an element, representing the number of protons in the nucleus. It uniquely identifies the element and plays a significant role in determining the element's chemical properties.
When discussing hydrogen-like ions, the atomic number \( Z \) influences the energy levels of the electron since the energy is dependent on how strongly the nucleus holds onto that single electron.
While \( Z \) affects the depth of energy levels, the formula \( E_n = -\frac{Z^2 \times 13.6 \text{ eV}}{n^2} \) shows that as \( Z \) increases, energy levels of the ion become more negative, or deeper. This implies that electrons are bound more tightly to the nucleus, thus needing more energy to be removed. However, \( Z \) does not affect the degeneracy, as it does not enter the equation \( g_n = n^2 \), which is solely dependent on \( n \).
  • The deeper the energy level (more negative), the harder it is to ionize the electron from the atom.
  • Changes in \( Z \) alter energy level depth but not the number of states sharing each energy level (degeneracy).

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Most popular questions from this chapter

(a) Calculate the expected harmonic-oscillator frequency of vibration for carbon monoxide, \(\mathrm{CO}\), if the force constant is \(1902 \mathrm{~N} / \mathrm{m}\). (b) What is the expected frequency of \({ }^{13} \mathrm{CO}\), assuming the force constant remains the same?

Construct an energy level diagram showing all orbitals for the hydrogen atom up to \(n=5\), labeling each orbital with its appropriate quantum numbers. How many different orbitals are in each shell?

Evaluate \(\langle r\rangle\) for \(\Psi_{1 s}\) (assume that the operator \(\hat{r}\) is defined as "multiplication by the coordinate \(r^{\prime \prime}\) ). Why does \(\langle r\rangle\) not equal \(0.529 \AA{A}\) for \(\Psi_{1 s}\) ? In this case, \(d \tau=4 \pi r^{2} d r\).

A harmonic oscillator has a frequency of \(7.04 \mathrm{~s}^{-1}\) and a force constant of \(866 \mathrm{~N} / \mathrm{m}\). What is the mass of the oscillator?

Simply using arguments based on odd or even functions, determine whether the following integrals involving harmonic oscillator wavefunctions are identically zero, are not identically zero, or are indeterminate. If indeterminate, state why. (a) \(\int_{-\infty}^{+\infty} \Psi_{1}^{*} \Psi_{2} d x\) (b) \(\int_{-\infty}^{+\infty} \Psi_{1}^{*} \bar{x} \Psi_{1} d x\) (c) \(\int_{-\infty}^{+\infty} \Psi_{1}^{*} \bar{x}^{2} \Psi_{1} d x\), where \(\hat{x}^{2}=\hat{x} \cdot \hat{x}\) (d) \(\int_{-\infty}^{+x} \Psi_{1}^{*} \Psi_{3} d x\) (e) \(\int_{-\infty}^{+x} \Psi_{3}^{+} \Psi_{3} d x\) (f) \(\int_{-\infty}^{+\infty} \Psi_{1}^{*} \hat{V} \Psi_{1} d x\), where \(\hat{V}\) is some undefined potential energy function.
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