Which of the following sets of quantum numbers is not allowed?
- $n =3,1=2, m _{ l }=0, s =+\frac{1}{2}$
- $n =3,1=2, m _1=-2, s =+\frac{1}{2}$
- $n =3,1=3, m _{ l }=-3, s =-\frac{1}{2}$
- $n =3,1=0, m _1=0, s =-\frac{1}{2}$
The Correct Option is C
Solution and Explanation
To determine which set of quantum numbers is not allowed, let's review the rules for quantum numbers:
- Principal quantum number (\(n\)): It must be a positive integer (1, 2, 3, ...).
- Azimuthal quantum number (\(l\)): It defines the shape of the orbital and ranges from 0 to \(n-1\).
- Magnetic quantum number (\(m_l\)): Its values range from \(-l\) to \(+l\), including zero.
- Spin quantum number (\(s\)): It can either be \(+\frac{1}{2}\) or \(-\frac{1}{2}\).
Now, let's analyze each option:
- Option 1: \(n = 3, l = 2, m_l = 0, s = +\frac{1}{2}\)
- All quantum numbers are valid: \(n = 3\) is a positive integer, \(l = 2\) is within [0, 2], \(m_l = 0\) is valid under \([-2, 2]\), and \(s = +\frac{1}{2}\).
- Option 2: \(n = 3, l = 2, m_l = -2, s = +\frac{1}{2}\)
- This is valid: \(m_l = -2\) lies in the range \([-l, l]\), i.e., [-2, 2] for \(l = 2\).
- Option 3: \(n = 3, l = 3, m_l = -3, s = -\frac{1}{2}\)
- This set is not allowed: For \(n = 3\), the maximum value of \(l\) is 2 (since \(l\) must be \(&leq n-1\)). Here \(l = 3\) is beyond the permissible range.
- Option 4: \(n = 3, l = 0, m_l = 0, s = -\frac{1}{2}\)
- This is valid: All values conform to the constraints (\(l = 0\) allows only \(m_l = 0\)).
Thus, the correct answer is: \(n = 3, l = 3, m_l = -3, s = -\frac{1}{2}\), as it does not conform to the quantum number rules, particularly violating the rule for \(l\).
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Concepts Used:
Quantum Mechanical Model of the Atom
Quantum Mechanics:
Quantum mechanics is an evolving and much-advanced field of science that aims at understanding the properties of matter and objects in relation to their corresponding atomic and sub-atomic nature. It further illustrates the characteristics of the atoms, protons, electrons, and neutrons specifically and in the context of each other. It aims at studying electromagnetic radiation as well. This is a sub-part of the wider theory of quantum physics.
Read Also: Quantum Mechanical Model of Atom
Quantum Mechanical Models:
Presently, the scientific world has only two acceptable and working models of quantum mechanics. Such as,
- The first model for the understanding and application of quantum mechanics that is acceptable currently is the Bohr Model.
The basis of this model of the Bohr is seen in terms of mathematics which is used for understanding the complex structures.
- Another acceptable model is the Quantum Mechanics Model which has its basis in quantum theory.
This quantum theory ultimately defines the exact properties of matter over a period of time. It usually works on the uncertainty principle.