To understand and be able to use the rules for determining a

To understand and be able to use the rules for determining allowable orbital angular momentum states. Several numbers are necessary to describe the states available to an electron in the hydrogen atom. The principal quantum number n determines the energy of the electron. The orbital quantum number l determines the total angular momentum of the electron, and the magnetic quantum number ml determines the component of the angular momentum parallel to a specific axis, usually the z axis. For a given principal quantum number n, the orbital quantum number can take integer values ranging from zero to n1. For a given orbital quantum number l, the magnetic quantum number can take integer values from l to l. A fourth number, the spin ms, is important for interactions with magnetic fields and counting states. The spin can be either +1/2 or 1/2, independent of the values of the other quantum numbers. The energy of an electron in hydrogen is related to the principal quantum number by En=(13.60eV)/n2. The orbital angular momentum is related to the orbital quantum number by L=l(l+1), and the orbital angular momentum in the z direction is related to the magnetic quantum number by Lz=ml.

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 = 3?

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

Solution

For State n=3

The possible values of orbital quantum numbers is l=0,1,2

For l=0, the possible values of m₁ =0, total values=0

For l=1, the possible values of m₁=-1,0,+1, total values =3

For l=2, the possible values of m₁=-2,-1,0,+1,+2, total values=5

n=3

l=0, m₁=0, m_s=+1/2, -1/2

n=3

l=0, m₁₌-2, m_s=+1/2,-1/2

l=0, m₁=-1, m_s=+1/2,-1/2

l=0, m₁=0, m_s=+1/2,-1/2

l=0, m₁=+1, m_s=+1/2,-1/2

l=0, m₁=+2, m_s=+1/2,-1/2

The total number of allowed states are multiply by 2 to account is =8*2=16.

Total possible states are 16.