Question

What are$E$ and $L$ (as a multiple of$h$) of a hydrogen atom in the $6f$ state?

Short answer

The hydrogen atom in the $6f$state is$E_n=-0.3777\text{eV}$and$L=\sqrt{12}ħ$.

Step-by-step solution

Given Information

We need to find are $E$and $L$(as a multiple of $h$) of a hydrogen atom in the $6f$state.

Simplify

Finding value for $E$and $L$of a hydrogen atom in the $6f$state.

know that the hydrogen atom in the $6f$state is described by $n=6$an $l=3$.

Then,

$$\begin{gathered} E_n=\frac{-13.6\text{eV}}{n^2} \\ {E}_n=\frac{-13.6\text{eV}}{6^2} \\ {E}_n=\frac{-13.6\text{eV}}{36} \\ {E}_n=-0.3777\text{eV} \end{gathered}$$

Now, finding the value for angular momentum $L$:

$$\begin{gathered} L=\sqrt{l\left(l+1\right)}ħ \\ L=\sqrt{3\left(3+1\right)}ħ \\ L=\sqrt{3\left(4\right)}ħ \\ L=\sqrt{12}ħ \end{gathered}$$