Q.Which of the following sets of quantum numbers are correct? Each set gives values of n, l and m_l respectively. (Note: more than one of the given options may be correct.)

The allowed values for the principal quantum number ($n$), azimuthal quantum number ($l$), and magnetic quantum number ($m_l$) are governed by specific rules derived from the quantization of energy levels in atoms.

Concept: Energy Level Quantization

1. The principal quantum number $n$ can be any positive integer ($n = 1, 2, 3, \dots$).
2. The azimuthal quantum number $l$ can take integer values from $0$ to $n-1$ ($0 \le l \le n-1$).
3. The magnetic quantum number $m_l$ can take integer values from $-l$ to $+l$, including $0$ ($-l \le m_l \le +l$).

Let's check each set:

• (A) $n = 1, l = 1, m_l = +2$: Invalid, because for $n=1$, $l$ can only be $0$. Here $l=1$, which violates $l \le n-1$.
• (B) $n = 2, l = 1, m_l = +1$: Valid. For $n=2$, $l$ can be $0, 1$. Here $l=1$ is allowed. For $l=1$, $m_l$ can be $-1, 0, +1$. Here $m_l=+1$ is allowed. …

Understanding Energy Level Quantization

In an atom, electrons do not orbit the nucleus in arbitrary paths or with arbitrary energies. Instead, their energy levels and spatial distributions are quantized, meaning they can only exist in specific, discrete states. These states are described by a set of four quantum numbers: the principal quantum number ($n$), the azimuthal (or angular momentum) quantum number ($l$), the magnetic quantum number ($m_l$), and the spin quantum number ($m_s$). This problem focuses on the first three, which describe the orbital an electron occupies.

These quantum numbers arise naturally from solving the Schrödinger equation for an electron in a hydrogen atom. Each quantum number has a specific role in defining the electron's state:

• Principal Quantum Number ($n$): Primarily determines the electron's energy level and the average distance of the electron from the nucleus (the size of the orbital). Higher $n$ means higher energy and a larger orbital.
• Azimuthal Quantum Number ($l$): Determines the shape of the electron's orbital and its angular momentum. It defines the subshell within a given principal shell.
• Magnetic Quantum Number ($m_l$): Determines the orientation of the orbital in space. It describes how the orbital is aligned when an external magnetic field is applied.

The key to solving this problem lies in understanding the allowed values for each quantum number and their interdependencies.

Rules for Quantum Numbers

The allowed values for $n$, $l$, and $m_l$ are governed by specific rules:

1. Principal Quantum Number ($n$):

   • $n$ can be any positive integer: $n = 1, 2, 3, \dots$.
   • It corresponds to the electron shell (K, L, M, etc.).

2. Azimuthal Quantum Number ($l$):

   • $l$ can take integer values from $0$ up to $n-1$.
   • So, $0 \le l \le n-1$.
   • The value of $l$ defines the type of subshell:
     • $l=0$ corresponds to an s-subshell (spherical shape).
     • $l=1$ corresponds to a p-subshell (dumbbell shape).
     • $l=2$ corresponds to a d-subshell (more complex shapes).
     • $l=3$ corresponds to an f-subshell.

3. Magnetic Quantum Number ($m_l$):

   • $m_l$ can take any integer value from $-l$ to $+l$, including $0$.
   • So, $-l \le m_l \le +l$.
   • For a given $l$, there are $2l+1$ possible values of $m_l$, which correspond to the number of orbitals in that subshell.

Step-by-Step Analysis of Each Option

Let's apply these rules to each given set of quantum numbers.

1. Option (A): $n = 1, l = 1, m_l = +2$

   • Check $l$: For $n=1$, the allowed values for $l$ are $0$ (since $l \le n-1 \implies l \le 1-1 \implies l \le 0$).
   • The given $l=1$ violates this rule.
   • Check $m_l$: Even if $l=1$ were allowed, the allowed values for $m_l$ would be $-1, 0, +1$ (since $-l \le m_l \le +l \implies -1 \le m_l \le +1$).
   • The given $m_l=+2$ violates this rule.
   • Conclusion: This set of quantum numbers is incorrect.

2. Option (B): $n = 2, l = 1, m_l = +1$ …