10. (a) Let the quantum numbers of the pair in question be n and n + 1, respectively. Then En+1 - En =
E1(n +1)2 - E1n2 =(2n +1)E1. Letting
En+1 - En =(2n +1)E1 =3(E4 - E3) =3(42E1 - 32E1) =21E1 ,
we get 2n + 1 = 21, or n = 10.
(b) Now letting
En+1 - En =(2n +1)E1 =2(E4 - E3) =2(42E1 - 32E1) =14E1 ,
we get 2n + 1 = 14, which does not have an integer-valued solution. So it is impossible to find the
pair of energy levels that fits the requirement.