Chapter 3: Problem 59
What is the significance of \(\psi_{210}\). Find out angular momentum, spherical nodes and angular node for \(\psi_{210}\).
Short Answer
Expert verified
For \(\psi_{210}\), the angular momentum is \(\sqrt{2}\hbar\), there are 0 spherical nodes, and 1 angular node.
Step by step solution
01
Understanding Quantum Numbers
The notation \(\psi_{nlm}\) refers to the wave function of an electron in a hydrogen atom where \(n\) is the principal quantum number, \(l\) is the angular momentum quantum number, and \(m\) is the magnetic quantum number. For \(\psi_{210}\), \(n=2\), \(l=1\), and \(m=0\).
02
Calculating Angular Momentum
Angular momentum \(L\) in quantum mechanics is given by \(L=\sqrt{l(l+1)}\hbar\), where \(\hbar\) is the reduced Planck constant. Since \(l=1\) for \(\psi_{210}\), the angular momentum \(L=\sqrt{1(1+1)}\hbar=\sqrt{2}\hbar\).
03
Determining Spherical Nodes
The number of spherical (radial) nodes is given by \(n - l - 1\). For \(\psi_{210}\), \(n=2\) and \(l=1\), so the number of spherical nodes is \(2 - 1 - 1 = 0\).
04
Identifying Angular Nodes
The number of angular nodes is equal to the angular momentum quantum number \(l\). For \(\psi_{210}\), \(l=1\), thus there is 1 angular node.
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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
Quantum mechanics introduces a different perspective on the concept of angular momentum compared to classical mechanics. In the subatomic world, angular momentum is quantized, meaning that it can only take on certain discrete values. The angular momentum of an electron orbiting an atom is an intrinsic property connected with its wave-like behavior.
For any electron in an atom, the angular momentum is determined by its angular momentum quantum number, denoted as 'l'. The allowed values for 'l' range from 0 up to one less than the principal quantum number 'n'. In the context of \(\psi_{210}\), the angular momentum quantum number 'l' is equal to 1. Using the formula \(L=\sqrt{l(l+1)}\hbar\), where \(\hbar\) is the reduced Planck constant, we calculate the angular momentum of the electron. In our case, \(L=\sqrt{2}\hbar\) signifies the quantized angular momentum for the given wave function. This quantization underpins many physical phenomena, such as the emission spectra of atoms.
For any electron in an atom, the angular momentum is determined by its angular momentum quantum number, denoted as 'l'. The allowed values for 'l' range from 0 up to one less than the principal quantum number 'n'. In the context of \(\psi_{210}\), the angular momentum quantum number 'l' is equal to 1. Using the formula \(L=\sqrt{l(l+1)}\hbar\), where \(\hbar\) is the reduced Planck constant, we calculate the angular momentum of the electron. In our case, \(L=\sqrt{2}\hbar\) signifies the quantized angular momentum for the given wave function. This quantization underpins many physical phenomena, such as the emission spectra of atoms.
Spherical Nodes
When dealing with the wave functions \(\psi_{nlm}\), those functions predict regions in space around an atom where the probability of finding an electron is precisely zero. These regions are called nodes, and they come in two varieties: spherical (radial) and angular nodes. Spherical nodes are related to the radial part of the wave function and exist in the spaces between the electron shells.
In the concept of the wave function \(\psi_{210}\), the formula to determine the number of spherical nodes is \(n - l - 1\). Here, 'n' represents the principal quantum number, which indicates the electron shell or energy level, and 'l' is the angular momentum quantum number. With \(n=2\) and \(l=1\), we can deduce there are 0 spherical nodes. This means there are no radial regions within this electron shell of the atom where the probability of finding the electron is zero. This detail is vital in understanding electron configurations and the chemical properties they influence.
In the concept of the wave function \(\psi_{210}\), the formula to determine the number of spherical nodes is \(n - l - 1\). Here, 'n' represents the principal quantum number, which indicates the electron shell or energy level, and 'l' is the angular momentum quantum number. With \(n=2\) and \(l=1\), we can deduce there are 0 spherical nodes. This means there are no radial regions within this electron shell of the atom where the probability of finding the electron is zero. This detail is vital in understanding electron configurations and the chemical properties they influence.
Angular Nodes
Angular nodes, also known as angular or azimuthal nodes, are regions where there is a zero probability of finding an electron, and they are determined by the wave function's angular part. The number of angular nodes is directly given by the angular momentum quantum number \(l\). These nodes correspond to planes or axes through the nucleus.
For our expression \(\psi_{210}\), the angular momentum quantum number 'l' equals 1, indicating that there is 1 angular node. This singular node can be visualized as a flat plane through the nucleus where the probability of finding the electron is nil. The presence and position of angular nodes affect the shape of the electron orbitals and help us understand the spatial distribution of electrons in an atom. As we delve into molecular bonding and geometries, recognizing how these nodes shape electron behavior is critical.
For our expression \(\psi_{210}\), the angular momentum quantum number 'l' equals 1, indicating that there is 1 angular node. This singular node can be visualized as a flat plane through the nucleus where the probability of finding the electron is nil. The presence and position of angular nodes affect the shape of the electron orbitals and help us understand the spatial distribution of electrons in an atom. As we delve into molecular bonding and geometries, recognizing how these nodes shape electron behavior is critical.