Show that the number of states with the same quantum number nis $\mathbf{2}{\mathbf{n}}^{\mathbf{2}}$.

The quantum number is n given.

The set of numbers used to describe the position and power of an electron atom is called quantum numbers. There are four quantum numbers, namely, prime numbers, azimuthal, magnetic numbers, and spin quantum. The stored values of the quantum system are given by quantum numbers.

For every value of the principal quantum number, there are n values of I ranging from 0 to (n-1). For every value of the orbital quantum number, there exist (2I+1) values of magnetic quantum numbers ranging from -I to +I. Thus, the maximum value of this magnetic quantum number is equal to the value of the orbital quantum number. Now, every electron has two spin orientations, thus the value of the magnetic quantum number is determined by these spins.

Formulae:

The number of possible electron states for each value of is,

$N_l=2(2l+1)$ ….. (1)

The total number of possible electron states for a given state is,

${\mathrm{N}}_{\mathrm{n}}=\sum _1^{\mathrm{n}-1}{\mathrm{N}}_{\mathrm{I}}$ ….. (2)

Now, using the concept, the total number of electron states can be given using equation (1) in equation (2) as follow.

$$\begin{gathered} N_n=\sum _1^{n-1}2\left(2I+1\right) \\ =\sum _1^{n-1}2\left(2I\right)+\sum _1^{n-1}2\left(1\right) \end{gathered}$$

$\mathrm{N}=4\sum _1^{\mathrm{n}-1}\left(\mathrm{I}\right)+2\sum _1^{\mathrm{n}-1}\left(1\right)$ ….. (3)

Now, solving both the terms individually, the value from the first term is as follow.

$4\sum _1^{n-1}\left(I\right)=4\frac{n}{2}\left(n-1\right)$

$4\sum _1^{n-1}\left(I\right)=2n^2-2n$ ….. (4)

Now, the value of the second term is given as:

$2\sum _1^{n-1}\left(1\right)=2n$ ….. (5)

Since, there are terms and the average terms is (n-1)./2

Now, substituting the values of equation (4) and (5) in equation (3), the required value of the electrons states is as follow.

$$\begin{gathered} N_n=2n^2-2n+2n \\ =2n^2 \end{gathered}$$

Hence, the number of the electron states is $2n^2$.