Question: Calculate the energy change required for an electron to move between states: a quantum jump up or down an energy-level diagram.

Electrons of an atom always revolve following certain paths around the nucleus called orbits or call these Electron Shells. Each orbit has its own energy-specific energy levels. The orbit closer to the nucleus has a low energy level, and the orbit far away from the nucleus has high energy levels.

The principal Quantum number $\left(n\right)$can be represented as the specific orbit number belonging to a particular electron state.

There is a relationship between the energy level $\left(E\right)$of the electron and the closeness of the orbit to the nucleus that is given by,

$E=-\frac{13.6}{n^2}{{\text{Z}}_{eff}}^2\text{eV}$ ….. (1)

Here, $Z$ is the atomic number.

Observe here that the energy level is negative with increasing $n$, which should always be a positive integer, i.e. $\left(n=1,\text{2, 3,}...\text{, n}\right)$. Hence the values of $E$will become as,

$$\begin{gathered} E_1=-13.6\text{eV} \\ {E}_2=-3.39\text{eV} \\ {E}_3=-1.51\text{eV}... \end{gathered}$$

Hence, the energy level is quantized.

Consider a Hydrogen atom that has an atomic number $\left(Z=1\right)$. The electron present in orbit with the quantum number $\left(n_1\right)$, which has the energy level $\left(E_1\right)$, gains energy to move to the next orbit with the quantum number $\left(n_2\right)$, which has the energy level $\left(E_2\right)$.

Then, the change in energy can be written from the equation (1) as:

$\begin{array}{rcl}\Delta E& =& E_2-E_1\\ & =& -\frac{13.6}{n_2^2}{Z}_{eff}^2\text{eV}-\left(-\frac{13.6}{n_1^2}{Z}_{eff}^2\text{eV}\right)\\ & =& 13.6\left(\frac{1}{n_1^2}-\frac{1}{n_2^2}\right)Z_{eff}^2\text{eV}\end{array}$

For elements other than Hydrogen, we can write equation (1) as,

$\Delta E=13.6\left(\frac{1}{n_1^2}-\frac{1}{n_2^2}\right)Z_{eff}^2\text{eV}$