Question

In Table 7.5, the pattern that develops with increasing n suggests that the number of different sets of$\left(\mathcal{l},m_{\mathcal{l}}\right)$values for a given energy level n is$n^2$. Prove this mathematically by summing the allowed values of$m_{\mathcal{l}}$for a given$\mathcal{l}$over the allowed values of$\mathcal{l}$for a given n.

Short answer

The number of different sets of$\left(l,m_l\right)$ values for a given energy level n is$n^2$ .

Step-by-step solution

 Given data

Energy level = n.

 Approach to find the final answer

For each n that is the principal quantum number, l, which is the azimuthal quantum number, can be from 0 to n – 1; for each n, there are 2l+1 allowed values of$m_l$.

 To prove the number of different sets of l,ml values for a given energy level n is n2

Summing all the allowed values of$m_l$ for a given l over the allowed values of l for a given n as:

$$\begin{gathered} \sum _0^{n-1}2l+1=2\left(\frac{0+n-1}{2}\right)n+n \\ =\left(n-1\right)n+n \\ =n^2-n+n \\ =n^2 \end{gathered}$$

Thus, the number of different sets of$\left(l,m_l\right)$ values for a given energy level n is$n^2$ .