Question

The quantum state of a particle can be specified by giving a complete set of quantum numbers (n, l, m_l, m_s). How many different quantum states are possible if the principal quantum number is n = 4?  

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_1 for each orbital quantum number. Sum these quantities, and then multiply by 2 to account for the two possible orientations of spin.

Answer 1

1)n= principal quantum number

Here n=4

2)l= orbital angular momentum or Azimuthal quantum number.

It takes values 0 to n-1

i.e 0,1,2,3 (s,p,d,f respectively)

3) = magnetic orbital quantum number.

It takes values : -l to +l

So for l=0 = 0

for l= 1 = -1,0,+1

  for l= 2 = -2,-1,0,+1,+2

for l= 3 = -3,-2,-1,0,+1,+2,+3

Total for values are 16

4) = magnetic spin quantum number

For each individual spin value mit has two values namely spin up and spin down, so it takes two values

Hence total spin values are 32

32 quantum states are possible for n=4