Answer to Question #279263 in Atomic and Nuclear Physics for Elie

Question

Using the concept of QM you have learned (application of the SE

to H-like atoms), derive / discuss the (existence of ) quantum

numbers n, l, ml

Answer 1

The three quantum numbers principal (n), azimuthal (l) and magnetic $\left(m_{l}\right)$ are interdependent on one another. This interdependence of quantum numbers can be shown as follows:

1. The principal quantum number [n] can have only positive integer values. So the possible values for principal quantum number are n=1,2,3... so on.

2. The azimuthal quantum number [l] can have its values from 0 to n-1. So the possible values for azimuthal quantum number are $l=0,1,2 \ldots[n-1].$

3. The magnetic quantum number $\left[m_{l}\right]$ can have its values varying from -1 to +1 . So the possible values for magnetic quantum number are $m_{t}=-l \ to +l.$

Thus on knowing the value of principal quantum number (n) we can first calculate the possible values of azimuthal quantum number (l) from the value of (l) we can calculate the corresponding values of magnetic quantum number $\left(m_{l}\right)$ .

For example, if we take, n=2

Then possible values of / are follows:

$\begin{aligned} &l=0 \text { to }[n-1] \\ &l=0,1 \end{aligned}$

And for this / the corresponding values of $m_{l}$ are as follows:

$\begin{aligned} &m_{t}=-l \ t o+l \\ &m_{l}=-2,-1,0,+1,+2 \end{aligned}$

Thus it is clearly shown that the value of $m_{1}$ depends on / which further depends on the value of n . In other words, the numerical values of quantum number n, I and m, are interdependent on each other.