Problem 63 - Fundamentals of Physics

Question

(a) For a given value of the principal quantum number \(n\) for a hydrogen atom, how many values of the orbital quantum number \(\ell\) are possible? (b) For a given value of \(\ell,\) how many values of the orbital magnetic quantum number \(m_{\ell}\) are possible? (c) For a given value of \(n\), how many values of \(m_{\ell}\) are possible?

Step-by-Step Solution

Answer

(a) n values; (b) 2ℓ + 1 values; (c) n^2 values.

Step 1: Understanding the Orbital Quantum Number (ℓ)

The orbital quantum number \( \ell \) for a given principal quantum number \( n \) can take integer values starting from 0 up to \( n-1 \). Thus, for a hydrogen atom, the number of possible values of \( \ell \) is \( n \).

Step 2: Evaluating Possible Values of Magnetic Quantum Number (m_ℓ) for Given ℓ

For each orbit angular momentum quantum number \( \ell \), the magnetic quantum number \( m_{\ell} \) can take values ranging from \( -\ell \) to \( +\ell \). Hence, there are \( 2\ell + 1 \) possible values for \( m_{\ell} \).

Step 3: Calculating Total Values of Magnetic Quantum Number (m_ℓ) for Given n

For a given \( n \), sum the values of \( m_{\ell} \) over all possible \( \ell \) values. \( \ell \) ranges from 0 to \( n-1 \), therefore sum the series \( 2\ell + 1 \) from \( \ell = 0 \) to \( \ell = n-1 \). This sums up to \( n^2 \) values. This result can also be confirmed by recognizing that each atomic orbital defines one set of \( m_{\ell} \), and there are \( n^2 \) such orbitals per principal quantum number.

Key Concepts

hydrogen atom

The hydrogen atom is one of the simplest and most fundamental systems studied in quantum mechanics. It consists of a single proton in the nucleus with an electron orbiting around it. Despite its simplicity, the hydrogen atom is crucial for understanding the quantum world, as it helps illustrate the behavior of electrons in atoms. Quantum mechanics describes the hydrogen atom by using quantum numbers, which define the electron's energy, angular momentum, and orientation.
The energy levels of the hydrogen atom are determined by the principal quantum number, symbolized as \( n \). The behavior and properties of the electron are further described by the orbital quantum number \( \ell \), magnetic quantum number \( m_{\ell} \), and spin quantum number \( m_s \). Together, these quantum numbers provide a comprehensive picture of the electron's state in the atom.

• Principal Quantum Number \( n \): Determines the energy level and size of the electron orbit.
• Orbital Quantum Number \( \ell \): Defines the shape of the electron's orbital.
• Magnetic Quantum Number \( m_{\ell} \): Dictates the orientation of the orbital in space.
• Spin Quantum Number \( m_s \): Specifies the electron's spin direction.

Understanding these quantum numbers is essential for grasping how an electron behaves in a hydrogen atom and how its properties change across different atoms.

orbital quantum number

The orbital quantum number denoted as \( \ell \) is one of the four quantum numbers used to describe the state of an electron in an atom. It is also known as the azimuthal quantum number. This quantum number plays a pivotal role in determining the shape of the electron's orbital.
For a given principal quantum number \( n \), the orbital quantum number \( \ell \) can take integer values ranging from 0 to \( n-1 \). Each value of \( \ell \) corresponds to a specific orbital type, namely:

• \( \ell = 0 \): s orbital (spherical shape)
• \( \ell = 1 \): p orbital (dumbbell shape)
• \( \ell = 2 \): d orbital (cloverleaf shape)
• \( \ell = 3 \): f orbital (complex shapes)

These shapes directly impact the distribution and behavior of the electron within an atom. Thus, knowing \( \ell \) helps predict the types of chemical bonds the atom can form and its chemical properties.
Understanding the orbital quantum number also allows for calculating allowable magnetic quantum numbers, \( m_{\ell} \), for each \( \ell \). Therefore, the orbital quantum number indirectly influences an electron's spatial orientation within a magnetic field, as described next.

magnetic quantum number

The magnetic quantum number, symbolized as \( m_{\ell} \), adds another layer of depth to our understanding of how electrons are situated within atoms. It defines the orientation of an electron's orbital relative to an external magnetic field.
For a given orbital quantum number \( \ell \), the magnetic quantum number \( m_{\ell} \) can take integer values ranging from \(-\ell\) to \(+\ell\). Therefore, there are \(2\ell + 1\) possible orientations for the electron's orbital. For example:

• \( \ell = 0 \): \( m_{\ell} = 0 \) (1 possible orientation)
• \( \ell = 1 \): \( m_{\ell} = -1, 0, 1 \) (3 possible orientations)
• \( \ell = 2 \): \( m_{\ell} = -2, -1, 0, 1, 2 \) (5 possible orientations)

This number dictates how the orbitals are arranged in space and how they can be filled by electrons. This becomes especially significant in atoms that interact with external fields or under electron quantum transitions.
It's important to see the magnetic quantum number as a key determinant in the distribution and pairing of electrons in multi-electron atoms. Thus, it influences both the chemical behavior of atoms and how they intermingle during chemical reactions.

principal quantum number

The principal quantum number \( n \) is one of the foundational quantum numbers in atomic theory. It signifies the energy level of an electron within an atom and is always a positive integer \( n = 1, 2, 3, \ldots \).
Each value of \( n \) dictates the size and energy of the electron cloud surrounding the nucleus. As \( n \) increases, the electron's probable distance from the nucleus advances, which also means higher energy levels. This quantum number gives rise to the concept of energy shells or layers around the nucleus.

• \( n = 1 \): First energy level (K shell)
• \( n = 2 \): Second energy level (L shell)
• \( n = 3 \): Third energy level (M shell)

Furthermore, the principal quantum number determines the number of possible values for other quantum numbers like the orbital quantum number \( \ell \). For a given \( n \), the number of possible orbital quantum numbers is \( n \), and for each \( \ell \), possible values of \( m_{\ell} \) are \( 2\ell + 1 \).
It's through this principal quantum number that we can comprehend how atoms with more than one electron are structured and how their electrons occupy distinct energy levels contributing to the atom's overall properties.