91Ó°ÊÓ

Each atom's electron is given one of four quantum numbers, the primary quantum number n , to describe the state of that electron.

A measurement of the angular momentum's magnitude is the orbital quantum number I .

The angular momentum vector's orientation in space is connected to the orbital magnetic quantum number${\mathbf{m}}_{\mathbf{I}}$ .

For a hydrogen atom, the values of orbital quantum number I for a given value of n are 0,1,2,....,(n-1) .

Hence, there are distinct values of orbital quantum number I are possible For a given value of the principal quantum number n for a hydrogen atom.

For a hydrogen atom, the values of orbital magnetic quantum number for a given value of I is $-\mathrm{I},-(\mathrm{I}-1),-(\mathrm{I}-2),............,+(\mathrm{I}-1),+\mathrm{I}$.

Hence, there are 2I+1 distinct values of orbital magnetic quantum number $m_I$for a given value of I .

In step 3, there are 2I+1 distinct values of orbital magnetic quantum number $m_I$for a given value of I .

Also, in step 2, there are n distinct values of orbital quantum number I are possible For a given value of the principal quantum number for a hydrogen atom.

The conclusion is that, there are 2n+1 distinct values of orbital magnetic quantum number $m_I$ for a given value of n .