The angle of orientation ($\theta$) of the angular momentum vector ($\vec{L}$) relative to the z-axis depends on the magnetic quantum number ($m_l$) and the orbital quantum number ($l$).

Angle Calculation Formula

The cosine of the angle $\theta$ between the angular momentum vector and the z-axis is given by the quantum mechanical relationship:

$\cos(\theta) = \frac{m_l}{\sqrt{l(l+1)}}$

Applying Values for H-atom State ($l=2$, $m_l=+2$)

We are given the state parameters for the H-atom: $l=2$ and $m_l=+2$. First, calculate the denominator term $\sqrt{l(l+1)}$:

$\sqrt{l(l+1)} = \sqrt{2(2+1)} = \sqrt{2 \times 3} = \sqrt{6}$

Next, substitute the quantum numbers into the cosine formula:

$\cos(\theta) = \frac{+2}{\sqrt{6}}$

Evaluate the fraction:

$\cos(\theta) \approx 0.81649658$

Finally, compute the angle $\theta$ by taking the inverse cosine (arccos):

$\theta = \arccos\left(\frac{2}{\sqrt{6}}\right) \approx \arccos(0.81649658)$

$\theta \approx 35.264 \text{ degrees}$

Final Result

Rounding the result to two decimal places as required, the angle of orientation is $35.26^\circ$. This value is consistent with the provided range check (33.3 to 36.9 degrees).