Q. A hydrogen-like atom (atomic number ) is in a higher excited state of quantum number . By successively emitting two photons of energies and respectively this excited atom can transition to the first excited state. By successively emitting two photons of energies and respectively the atom can also transition to the second excited state. If the value of is then find . (ionization energy of hydrogen atom )

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Answer: 6

Solution:

For the transition from a higher state n to the first excited state



For the eventual transition to the second excited
state
Thus,

Dividing equations (i) and (ii), we get


Substituting n=6 in any of the above equations,
we obtain