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Chapter 29: Problem 7

(a) How many different \(5 g\) states does hydrogen have? (b) Which of the states in part (a) has the largest angle between \(\overrightarrow{\boldsymbol{L}}\) and the \(z\) axis, and what is that angle? (c) Which of the states in part (a) has the smallest angle between \(\vec{L}\) and the \(z\) axis, and what is that angle?

Short Answer

Expert verified
(a) 9 different states. (b) Largest angle at \(\frac{\pi}{2}\). (c) Smallest angle at \(\approx 26.57^\circ\).

Step by step solution

01

Understanding Quantum Numbers

In hydrogen, the principal quantum number \( n \) describes the energy level, and the azimuthal quantum number \( l \) describes the orbital angular momentum. For a 'g' state, \( l = 4 \). The magnetic quantum number \( m_l \) can vary from \( -l \) to \( l \).
02

Determine Number of 5g States

Each value of \( m_l \) corresponds to a different state of the 5g orbital. Since \( m_l \) ranges from \(-4\) to \(4\), the possible values are \(-4, -3, -2, -1, 0, 1, 2, 3, 4\). Hence, there are 9 different 5g states.
03

Calculating the Angle - General Expression

The angle \( \theta \) between \( \vec{L} \) and the \( z \) axis is given by \( \cos\theta = \frac{m_l}{\sqrt{l(l+1)}} \). This formula will be used to find both the largest and smallest angles.
04

Finding the State with Largest Angle

The largest angle corresponds to the smallest \(|m_l|\), which in this case is \(m_l = 0\). The angle \( \theta \) is \( \cos^{-1}(0) = \frac{\pi}{2} \).
05

Finding the State with Smallest Angle

The smallest angle is when \( |m_l| \) is maximum, which occurs at \(|m_l| = 4\). The angle \( \theta \) is calculated as \[ \cos^{-1}\left(\frac{4}{\sqrt{4 \times 5}}\right) = \cos^{-1}\left(\frac{4}{\sqrt{20}}\right) = \cos^{-1}\left(\frac{4}{\sqrt{20}}\right) = \cos^{-1}\left(\frac{2}{\sqrt{5}}\right) \]. Calculating this gives \( \theta \approx 26.57^\circ \).

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Key Concepts

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

Quantum Numbers
Understanding quantum numbers is fundamental in quantum mechanics, as they describe various properties of electrons in an atom. Each electron is characterized by a set of four quantum numbers:
  • Principal Quantum Number ( \( n \) ): Indicates the size and energy level of an electron's orbit. It can take any positive integer value (1, 2, 3,...).
  • Azimuthal Quantum Number ( \( l \) ): Sometimes called the angular momentum quantum number, it determines the shape of an orbital. The value of \( l \) ranges from 0 to \( n-1 \).
  • Magnetic Quantum Number ( \( m_l \) ): Specifies the orientation of the orbital in space and ranges from \( -l \) to \( +l \).
  • Spin Quantum Number on: Associated with the intrinsic spin of the particle, it can have values of \( +\frac{1}{2} \) or \( -\frac{1}{2} \).
For a 5g state, the principal quantum number \( n \) is 5, and the azimuthal quantum number \( l \) is 4, as 'g' corresponds to \( l = 4 \). The magnetic quantum number \( m_l \) characterizes the different orientations possible for this orbital.
Angular Momentum
Angular momentum in quantum mechanics, denoted by \( \vec{L} \), plays a critical role in the quantum behavior of electrons. It is quantized and can only take specific values determined by quantum numbers. Here, the key aspects of angular momentum include:
  • The magnitude of the angular momentum vector is given by \( L = \sqrt{l(l+1)} \hbar \), where \( l \) is the azimuthal quantum number and \( \hbar \) is the reduced Planck's constant.
  • Angular momentum signifies the rotational motion, analogous to classical mechanics but with discretized values in quantum systems.
  • Angular momentum vectors have both a magnitude and an associated direction, with the orientation related to the magnetic quantum number \( m_l \).
The number of possible orientations, or states, for angular momentum vectors depend on the quantum number \( l \), with each distinct value of \( m_l \) representing a unique orientation for the angular momentum vector.
Magnetic Quantum Number
The magnetic quantum number \( m_l \) is an essential quantum number for defining the specific energy state of an electron within an atom. Here's what you need to know:
  • The range of \( m_l \) for a given azimuthal quantum number \( l \) is from \( -l \) to \( +l \), inclusive, which means there are \( 2l + 1 \) possible values for \( m_l \).
  • Each different value of \( m_l \) corresponds to a different orientation of the electron's angular momentum. For a 5g state, where \( l = 4 \), \( m_l \) can range from \(-4\) to \(4\). This gives us 9 possible states: \( -4, -3, -2, -1, 0, 1, 2, 3, 4 \).
  • The magnetic quantum number affects an electron's energy when it is placed in a magnetic field, although in the absence of a field (like in hydrogen), it primarily determines the direction of the electron鈥檚 angular momentum.
Thus, \( m_l \) plays a crucial role in determining the number of spatial orientations available to an electron's angular momentum vector, particularly important in understanding atomic spectra.
Angle Calculation in Quantum Mechanics
In quantum mechanics, the angle between the angular momentum vector \( \vec{L} \) and the z-axis is an important concept. This angle can be calculated using the formula:\[ \cos \theta = \frac{m_l}{\sqrt{l(l+1)}} \]
  • The angle \( \theta \) gives insight into the orientation of the angular momentum relative to a chosen axis, often the z-axis in quantum problems.
  • When \( m_l = 0 \), the angle \( \theta \) is \( 90^\circ \) or \( \pi/2 \) radians, indicating that \( \vec{L} \) is perpendicular to the z-axis. This orientation has the maximum angle possible.
  • For maximum \( |m_l| \), which is in a 5g state \( m_l = \pm 4 \), the angle \( \theta \) is smallest, calculated as \( \cos^{-1}(\frac{4}{\sqrt{20}}) \approx 26.57^\circ \).
  • This calculation helps predict electron behavior in different states and is crucial for visualizing atomic orbitals and their orientations in space.
Understanding these angles is key to exploring more complex systems where interactions with fields or other particles may alter typical angular distributions.

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Most popular questions from this chapter

The gap between valence and conduction bands in diamond is \(5.47 \mathrm{eV}\). (a) What is the maximum wavelength of a photon that can excite an electron from the top of the valence band into the conduction band? In what region of the electromagnetic spectrum does this photon lie? (b) Explain why pure diamond is transparent and colorless. (Hint: Will photons of visible light that strike a diamond be absorbed or transmitted?) (c) Most gem diamonds have a yellow color. Explain how impurities in the diamond can cause this color.

(a) List the different possible combinations of quantum numbers \(l\) and \(m_{l}\) for the \(n=5\) shell. (b) How many electrons can be placed in the \(n=5\) shell?

The dissociation energy of the hydrogen molecule (i.e., the energy required to separate the two atoms) is \(4.48 \mathrm{eV}\). In the gas phase (treated as an ideal gas), at what temperature is the average translational kinetic energy of a molecule equal to this energy?

The \(z\) component of the angular momentum of an atom, \(L_{z}\), is measured as a function of the angle that the angular momentum vector \(\vec{L}\) makes with the positive \(z\) axis, \(\theta .\) The resulting data are given in the table, where the angular momentum is measured in units of \(\hbar\). $$ \begin{array}{cc} \hline \boldsymbol{\theta}\left({ }^{\circ}\right) & \boldsymbol{L}_{z}(\hbar) \\\ \hline 63.7 & 1.97 \\ 79.2 & 1.05 \\ 90.4 & -0.03 \\ 102 & -0.95 \\ 118 & -2.02 \\ \hline \end{array} $$ Make a plot of \(L_{z}\) as a function of \(\cos \theta\). Using a linear "best fit" to the data, determine (a) the magnitude of the angular momentum vector in units of \(\hbar\) and (b) the corresponding angular momentum quantum number.

Use the electron configurations of He, Ne, and Ar to explain why these atoms normally do not combine chemically with other atoms.
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