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Chapter 7: Problem 77

What are the possible values for the quantum numbers \(n, \ell,\) and $m_{\ell} ?$

Short Answer

Expert verified
The possible values for the quantum numbers \(n\), \(\ell\), and \(m_{\ell}\) are: 1. \(n\) can be any positive integer (e.g. \(n = 1, 2, 3, \dots\)). 2. \(\ell\) ranges from \(0\) to \(n-1\) (e.g. if \(n = 3\), then \(\ell = 0, 1, 2\)). 3. \(m_{\ell}\) ranges from \(-\ell\) to \(+\ell\) (e.g. if \(\ell = 2\), then \(m_{\ell} = -2, -1, 0, 1, 2\)).

Step by step solution

01

Possible values for n (principal quantum number)

The principal quantum number, n, represents the energy level of an electron within an atom and can be any positive integer (n = 1, 2, 3, ...). There is no upper limit for n, as it can have any positive integer value.
02

Possible values for ℓ (azimuthal quantum number)

The azimuthal quantum number, ℓ, represents the shape of the electron's orbital and is dependent on the principal quantum number (n). The possible values for ℓ range from 0 to n-1. For example, if n = 1, then the possible values for ℓ would be: ℓ = 0. If n = 2, then the possible values for ℓ would be: ℓ = 0, 1. If n = 3, then the possible values for ℓ would be: ℓ = 0, 1, 2. And so on for higher values of n.
03

Possible values for mₗ (magnetic quantum number)

The magnetic quantum number, mₗ, represents the orientation of the electron's orbital and is dependent on the azimuthal quantum number (ℓ). The possible values for mₗ range from -ℓ to +ℓ. For example, if ℓ = 0, then the possible values for mₗ would be: mₗ = 0. If ℓ = 1, then the possible values for mₗ would be: mₗ = -1, 0, 1. If ℓ = 2, then the possible values for mₗ would be: mₗ = -2, -1, 0, 1, 2. And so on for higher values of ℓ. To summarize, the possible values for the quantum numbers n, ℓ, and mₗ are: 1. n = 1, 2, 3, ... 2. ℓ = 0, 1, 2, ... , n-1 3. mₗ = -ℓ, -(ℓ-1), ... , 0, ... , (ℓ-1), ℓ

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