Question

a. Draw a diagram similar to Figure 41.2 to show all the possible orientations of the angular momentum vector $\overrightarrow{L}$for the case $l=3$. Label each $\overrightarrow{L}$with the appropriate value of $m$.

b. What is the minimum angle between $\overrightarrow{L}$and the z-axis?

Short answer

Part (a) The diagram is

Part (b) the minimum angle between $\overrightarrow{L}\text{and}z\text{-axis is}30°$

Step-by-step solution

Step 1. Given information

(a) The number of $m$values for a particular value of $l$is given by the following equation$2l+1$

For $l=3$the values for$m\text{are:}-3,-2,-1,0,1,2,3$

The diagram showing all possible orientations of angular momentum vector for the case of $l=3$along with the appropriate $m$values for each orientation is given below:

Step 2. (b) The angle between the vector L→ and z-axis is

${\theta }_{lm}={\cos }^{-1}\left(\frac{m}{\sqrt{l(l+1)}}\right)$

${\theta }_{33}={\cos }^{-1}\left(\frac{3}{\sqrt{3(3+1)}}\right)$

$={\cos }^{-1}\left(\frac{3}{\sqrt{12}}\right)$

$={\cos }^{-1}(0.87)$

$=30°$

Thus, the minimum angle between $\overrightarrow{L}\text{and}z-\text{axis}$$=30°$