Answer
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Work Step by Step
The possible values for the quantum numbers \( n, \ell \), and \( m_{\ell} \) are determined by the rules of quantum mechanics. Here are the ranges for each quantum number:
1. Principal Quantum Number (\( n \)):
- \( n \) can take on positive integer values starting from 1 (1, 2, 3, ...).
2. Azimuthal Quantum Number (\( \ell \)):
- \( \ell \) can range from 0 to \( n-1 \), where \( n \) is the principal quantum number. So, \( \ell \) can be 0, 1, 2, ..., \( n-1 \).
3. Magnetic Quantum Number (\( m_{\ell} \)):
- \( m_{\ell} \) can take on integer values from \( -\ell \) to \( +\ell \). For example, if \( \ell = 2 \), then \( m_{\ell} \) can be -2, -1, 0, 1, or 2.
These quantum numbers describe the energy, shape, and orientation of atomic orbitals in quantum mechanics.
