Determine the number of enegry states in the hydrogen atom corresponding to the principle quantum number n` = 2` and calculate the energies of these states.
Strategy: An energy state of the hydrogen atom can be respresented by the wave function `psi_(n,l,m_(1))`, which can be specified by sunstituting the values of the three quantum numbers.

Explanation

For `n=2,l` can have the values `0` and `1`. For `l = 0, m_(l)` can only be `0`, corresponding to the state `psi_(2,0,0)`. For `l = 1, m_(l)` can have the values `-1,0`, or `1` corresponding to the states `psi_(2,1,-1),psi_(2,1,0)`, and `psi_(2,1,1)`.
Hence, we have four allowed states:
`{:(2s,,,,psi_(2,0,0)),(2p_(x),,,,psi_(2,1,-1)),(2p_(y),,,,psi_(2,1,0)),(2p_(z),,,,psi_(2,1,-1,)):}`
Since all of these states have the same principle quantum number `n = 2`, they also have the sam energy, which can be calculated using Eq. For `n =2`, we get
`E_(2) ==-(13.6)/((2)^(2))eV atom^(-1)`
`= -3.4eV "atom"^(-1)`