# Energy difference between the successive levels in same shell of atom

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Hydrogen-like atom is in an external magnetic field $B=1.6 \ T$.
a) What is the energy difference ($\triangle E$) between the successive $m_l$-levels in same $l$-shell of atom? (Give the answer in $eV$.)

I tried this:

$E_n = -\frac{13.6 \ eV}{n^2}$

$\triangle E = E_2 - E_1 = -\frac{13.6 \ eV}{2^2} - (-\frac{13.6 \ eV}{1^2}) = -\frac{13.6 \ eV}{4} + 13.6 \ eV =10.2 \ eV$

The question says the $m_l$ levels are in the same $l$ shell. This means that $n$ and$l$ are the same. You are using the wrong formula. You need a formula which involves magnetic field.
The splitting of the energies of the $m_l$ levels is called the Zeeman Effect. The splitting is $\Delta E=\mu B$ where $\mu$ is the Bohr magneton.