The quantum state of a particle can be specified by giving a complete set of quantum numbers (n, l, ml, ms). how many different quantum states are possible if the principal quantum number is n = 2?

To find the total number of allowed states, first, write down the allowed orbital quantum numbers l, and then write down the number of allowed values of m l for each orbital quantum number. Sum these quantities, and then multiply by 2 to account for the two possible orientations of spin.

Quantum states for n=5

l ... ml

0, 0 ... 1

1, - 1, 0, + 1 ... 3

2, - 2, - 1, 0, + 1, + 2 ... 5

3, - 3, - 2, - 1, 0, + 1, + 2, + 3 ... 7

4, - 4,-3, - 2, - 1, 0, + 1, + 2, + 3,+4 ... 9

2 * (1+3+5+7+9) =

2*5^2 = 50 allowed states