p40_010.pdf

10. (a) Let the quantum numbers of the pair in question be n and n + 1, respectively. Then E n +1 −

E n =

E 1 ( n +1) 2

−

E 1 n 2 =(2 n +1) E 1 . Letting

E n +1 −

E n =(2 n +1) E 1 =3( E 4 −

E 3 )=3(4 2 E 1 −

3 2 E 1 )=21 E 1 ,

we get 2 n + 1 = 21, or n = 10.

(b) Now letting

E n +1 − E n =(2 n +1) E 1 =2( E 4 − E 3 )=2(4 2 E 1 − 3 2 E 1 )=14 E 1 ,

we get 2 n + 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.