Question

The relation between n_m (n_m = the number of permissible values of magnetic quantum number (m)) for a given value of azimuthal quantum number (l), is

• A. l = 2n_m + 1
• B. n_m = 2l² + 1
• C. n_m = l + 2
• D. \(l=\frac {n_m-1}{2}\)

Answer 1

The Correct Option is D

To determine the relation between the number of permissible values for the magnetic quantum number \( n_m \) and a given azimuthal quantum number \( l \), let's delve into quantum mechanics.

Background

In atomic physics, the magnetic quantum number \( m \) describes the orientation of the orbital’s angular momentum in space relative to a magnetic field. The azimuthal quantum number \( l \) defines the shape of the orbital, and it can take integer values from 0 to \( n-1 \), where \( n \) is the principal quantum number.

Magnetic Quantum Number \( m \)

For a given azimuthal quantum number \( l \), the magnetic quantum number \( m \) can range from \(-l\) to \(+l\), including zero. This means:

\(m = -l, -l+1, \ldots, 0, \ldots, l-1, l\)

Therefore, the total number of permissible values for \( m \), which is \( n_m \), is calculated as:

\(n_m = 2l + 1\)

Explaining the Options

1. Option 1 (l = 2n_m + 1): Doesn't align with the basic definition.
2. Option 2 (n_m = 2l² + 1): Incorrect; the dependence on \( l^2 \) is not theoretically grounded.
3. Option 3 (n_m = l + 2): Incorrect; doesn't match the formula \( n_m = 2l + 1 \).
4. Option 4 (\(l = \frac {n_m-1}{2}\)): This is the correct relation derived from the formula \( n_m = 2l + 1 \).

Conclusion

The correct relation is given by Option 4: \(l = \frac {n_m-1}{2}\). This accurately describes how one can calculate the azimuthal quantum number \( l \) if the number of magnetic quantum numbers \( n_m \) is known.

Tip

Remember, the relation \( n_m = 2l + 1 \) is fundamental for understanding orbitals and how they split in magnetic fields, key concepts in quantum chemistry.