Chapter 28, Problem 19P

To determine

The possible values for $n,m_l,m_s$

Answer to Problem 19P

The principal quantum number: $n\ge 5$

The magnetic quantum number: $m_l=-4,-3,-2,-1,0,1,2,3,4$

The spin quantum number: $m_s=-\frac{1}{2},+\frac{1}{2}$

Explanation of Solution

Given info: orbital quantum number, $l=4$

Formula used:

The principal quantum number $n$ can take on any integer value $\left(1,2,3,\mathrm{...}\right)$ and also, corresponds to the quantum number of the old Bohr theory.

The orbital quantum number $l$ can take on values from $0$ to $\text{n-1}$ .

The magnetic quantum number, $m_l$ , can take on the integer values from $-l$ to $+l$

The spin quantum number, $m_s$ , can be $+\frac{1}{2}$ to $-\frac{1}{2}$

Calculation:

The value of $l$ can range from $\text{0}$ to $\text{n-1}$ .For $l=4$ : $n\ge 5$ .

For each $l$ the value of $m_l$ can range from $-l$ to $+l$ : $m_l=-4,-3,-2,-1,0,1,2,3,4$

Also, for each of these there are two values of $m_s$ : $m_s=-\frac{1}{2},+\frac{1}{2}$

Conclusion:

All the required values are calculated.