Chapter 41: Problem 15
a) How many different 5\(g\) states does hydrogen have? (b) Which of the states in part (a) has the largest angle between \(\vec L\) 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
There are 9 different 5g states. The largest angle is 90° and the smallest is approximately 26.57°.
Step by step solution
01
Understand the Quantum Numbers
For a hydrogen atom, the principal quantum number is denoted by \(n\), and the azimuthal or angular momentum quantum number is \(l\). The magnetic quantum number \(m_l\) ranges from \(-l\) to \(+l\). A 5\(g\) state indicates \(n = 5\) and \(l = 4\), since \(g\) corresponds to \(l = 4\).
02
Determine the Number of 5g States
The number of different magnetic states for a given \(l\) is determined by the number of possible \(m_l\) values, which are \(2l +1\). For \(l = 4\), the \(m_l\) values range from \(-4\) to \(+4\), giving us \(9\) possible states: \(-4, -3, -2, -1, 0, 1, 2, 3, 4\).
03
Use Vector Model of Angular Momentum
The angle \(\theta\) between \(\vec{L}\) (the angular momentum vector) and the z-axis is given by \(\cos{\theta} = \frac{m_l}{\sqrt{l(l+1)}}\). We will use this formula to find the angles for different \(m_l\) values.
04
Largest Angle Calculation
The largest angle occurs when \(|m_l|\) is smallest because the cosine value is smallest. For \(m_l = 0\), the angle \(\theta\) is \(\cos^{-1}(0) = \frac{\pi}{2} \text{ radians} = 90^{\circ}\).
05
Smallest Angle Calculation
The smallest angle occurs when \(|m_l|\) is largest, \(m_l = \pm4\). The angle is \(\theta = \cos^{-1}\left(\frac{4}{\sqrt{4(4+1)}}\right) = \cos^{-1}\left(\frac{4}{\sqrt{20}}\right) = \cos^{-1}\left(\frac{4}{2\sqrt{5}}\right)\). Simplifying gives \(\cos^{-1}\left(\frac{2}{\sqrt{5}}\right)\).
06
Calculate Smallest Angle Numerically
Compute \(\cos^{-1}(\frac{2}{\sqrt{5}})\): Use a calculator to find \(\cos^{-1}(0.894)\), which is approximately \(26.57^{\circ}\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Angular Momentum
In quantum mechanics, angular momentum is an intrinsic property of particles, such as electrons in an atom. It is analogous to the classical concept of angular momentum which describes the rotation of objects. However, in quantum systems, angular momentum is quantized, meaning it only takes on specific discrete values.
For atoms, the angular momentum of electrons is described using quantum numbers. These quantum numbers allow us to determine the energy level and shape of an electron's orbit. Angular momentum is an essential factor in understanding atomic structures and electron configurations.
For atoms, the angular momentum of electrons is described using quantum numbers. These quantum numbers allow us to determine the energy level and shape of an electron's orbit. Angular momentum is an essential factor in understanding atomic structures and electron configurations.
- The total angular momentum \(\vec L\) is a vector quantity combining both magnitude and direction.
- Magnitude is calculated as \(|\vec L| = \sqrt{l(l+1)}\hbar\), where \(\hbar\) is the reduced Planck's constant.
- Direction is often described with respect to a chosen axis, typically the z-axis in spherical coordinates.
Magnetic Quantum Number
The magnetic quantum number, denoted by \(m_l\), is one of the quantum numbers that describe an electron's state in an atom. It specifically pertains to the orientation of the angular momentum vector in the presence of an external magnetic field.
This quantum number can take values ranging from \(-l\) to \(+l\), where \(l\) is the azimuthal (or orbital angular momentum) quantum number. Each of these values corresponds to a different possible orientation of the electron cloud around the nucleus.
This quantum number can take values ranging from \(-l\) to \(+l\), where \(l\) is the azimuthal (or orbital angular momentum) quantum number. Each of these values corresponds to a different possible orientation of the electron cloud around the nucleus.
- For \(l = 4\) as in the 5\(g\) state, \(m_l\) values are \(-4, -3, -2, -1, 0, 1, 2, 3, 4\).
- The number of different states is given by \(2l + 1\).
Hydrogen Atom
The hydrogen atom is the simplest atom and serves as a fundamental model in quantum mechanics. It comprises one electron and one proton, which makes it ideal for studying the principles of atomic structure and the quantum mechanics governing electron behavior.
In the hydrogen atom, the electron moves under the influence of the electrostatic force exerted by the proton. The distinct energy levels and orbitals of the electron are described using various quantum numbers, which emerge from solving the Schrödinger equation for this system.
In the hydrogen atom, the electron moves under the influence of the electrostatic force exerted by the proton. The distinct energy levels and orbitals of the electron are described using various quantum numbers, which emerge from solving the Schrödinger equation for this system.
- It has a principal quantum number (\(n\)), which determines the main energy level of the electron.
- A hydrogen atom exhibits a spherical symmetry, allowing the usage of spherical coordinates \(r, \theta, \phi\) to describe electron position.
Azimuthal Quantum Number
The azimuthal quantum number, also known as the angular momentum quantum number and symbolized by \(l\), defines the shape of an electron's orbital. It is crucial in determining the spatial distribution of the electron cloud around the nucleus.
The values of \(l\) range from 0 to \(n-1\), where \(n\) is the principal quantum number. Each value of \(l\) corresponds to a different type of orbital, designated by letters (s, p, d, f, g, etc.). For a given principal quantum number, multiple possible orbital shapes are defined by the azimuthal quantum number.
The values of \(l\) range from 0 to \(n-1\), where \(n\) is the principal quantum number. Each value of \(l\) corresponds to a different type of orbital, designated by letters (s, p, d, f, g, etc.). For a given principal quantum number, multiple possible orbital shapes are defined by the azimuthal quantum number.
- \(l = 0\) corresponds to s-orbitals, \(l = 1\) to p-orbitals, and so on.
- For \(l = 4\), it is associated with the g-orbital, as seen in the 5\(g\) state.