Question

What are some examples of quantum numbers?

Answer 1

Hint: The state of an electron in an atom can be described by the set of four numbers that are solutions of the Schrodinger wave equation and are known as quantum numbers. These four quantum numbers are principal quantum number (n), azimuthal quantum number (l), magnetic quantum number $\left( {{\text{m}}_{\text{l}}} \right)$, and spin quantum number $\left( {{\text{m}}_{\text{s}}} \right)$.

Complete answer:
The values obtained from solving the Schrodinger wave equation that is acceptable by the wave equation for hydrogen atoms are known as quantum numbers. There are a total of four quantum numbers that describe the probable position and energy of an electron in an atom.
These four quantum numbers are briefly described below:
1 – Principal quantum number (n): It denotes the principal shell number and energy of an electron. Its value can be any positive integer. For example, for an electron present in a 2p subshell, the value of n is 2.
\[\text{n}=1,2,3.....\]
2 – Azimuthal quantum number (l): It denotes the subshell of an orbital within a specific principal quantum number in which an electron is present. It also specifies the shape of the orbital. The subshell with $\text{n}=3$and $l=0$ is the 3s subshell. A letter is used to identify each subshell.
\[\begin{align}
  & l\text{ 0 1 2 3 4 5 }.... \\
 & \text{Letter s p d f g h }.... \\
\end{align}\]
3 – Magnetic quantum number $\left( {{\text{m}}_{\text{l}}} \right)$ : It gives an insight into the spatial orientation of an orbital and its value is related to the value of the azimuthal quantum number. Its value range is given as:
\[{{\text{m}}_{l}}=-l,\text{ }\left( -l+1 \right),....,-1,0,1,....\left( l-1 \right),+l\]
4 – Spin quantum number $\left( {{\text{m}}_{\text{s}}} \right)$ : It tells the spin orientation of electrons in an orbital. An electron can spin in only one of two directions that are denoted as up spin $\left( \uparrow \right)$ and down spin $\left( \downarrow \right)$. Thus, there can be only two values possible:
\[{{\text{m}}_{\text{s}}}=+\dfrac{1}{2}\text{ }\left( \uparrow \right)\text{ or }-\dfrac{1}{2}\text{ }\left( \downarrow \right)\]
Hence, some examples of quantum numbers can be given as:

Orbital	Principal quantum number (n)	Azimuthal quantum number(l)	Magnetic quantum number$\left( {{\text{m}}_{\text{l}}} \right)$	Spin quantum number$\left( {{\text{m}}_{\text{s}}} \right)$
2p	2	1	-1, 0, +1	\[+\dfrac{1}{2}\text{ or }-\dfrac{1}{2}\]
4d	4	2	-2, -1, 0, +1, +2	\[+\dfrac{1}{2}\text{ or }-\dfrac{1}{2}\]

Note:
According to the Pauli exclusion principle, no two electrons can share the same set of four quantum numbers because an orbital cannot occupy more than two electrons. Also, for two electrons in the same orbital, the spins must be opposite to each other to minimize the electronic repulsion.