Question

When all quantum numbers are considered, how many different quantum states are there for a hydrogen atom with$n=1$? With$n=2$? With$n=3$? List the quantum numbers of each state.

Short answer

The quantum states are there for a hydrogen atom with $n=1$is $2$states,$n=2$is$8states$and$n=3$is$18$states.

Step-by-step solution

Given Information

We need to find different quantum states are there for a hydrogen atom with $n=1,n=2$and$n=3$.

Simplify

We finding a list of the quantum numbers of each state.
We know that, the two electrons have a set of quantum numbers$\left(n,l,m,m_s\right)$.

First, we find for $n=1$:

$\left(1,0,0,-\frac{1}{2}\right),\left(1,0,0,+\frac{1}{2}\right)$

Now, finding for $n=2$:

$$\begin{gathered} \left(2,0,0,-\frac{1}{2}\right),\left(2,0,0,+\frac{1}{2}\right),\left(2,1,-1,-\frac{1}{2}\right),\left(2,1,-1,+\frac{1}{2}\right) \\ \left(2,1,0,-\frac{1}{2}\right),\left(2,1,0,+\frac{1}{2}\right),\left(2,1,1,-\frac{1}{2}\right),\left(2,1,1,+\frac{1}{2}\right) \end{gathered}$$

Then, finding for $n=3$:

$$\begin{gathered} \left(3,0,0,-\frac{1}{2}\right),\left(3,0,0,+\frac{1}{2}\right),\left(3,1,-1,-\frac{1}{2}\right),\left(3,1,-1,+\frac{1}{2}\right),\left(3,1,0,-\frac{1}{2}\right),\left(3,1,0,+\frac{1}{2}\right) \\ \left(3,1,1,-\frac{1}{2}\right),\left(3,1,1,+\frac{1}{2}\right),\left(3,2,2,-\frac{1}{2}\right),\left(3,2,2,+\frac{1}{2}\right),\left(3,2,1,-\frac{1}{2}\right),\left(3,2,1,+\frac{1}{2}\right) \\ \left(3,2,0,-\frac{1}{2}\right),\left(3,2,0,+\frac{1}{2}\right),\left(3,2,-1,-\frac{1}{2}\right),\left(3,2,-1,+\frac{1}{2}\right),\left(3,2,-2,-\frac{1}{2}\right),(3,2,-2,+\frac{1}{2}) \end{gathered}$$